/*-*- compile-command: "/usr/bin/gcc -c -o ellQ2.gp.o -O3 -Wall -fno-strict-aliasing -fomit-frame-pointer -fPIC -I"/usr/local/include" ellQ2.gp.c && /usr/bin/gcc -o ellQ2.gp.so -shared -O3 -Wall -fno-strict-aliasing -fomit-frame-pointer -fPIC -Wl,-shared ellQ2.gp.o -lc -lm -L"/usr/local/lib" -lpari"; -*-*/ #include /* GP;install("init_ellQ2","v","init_ellQ2","./ellQ2.gp.so"); GP;install("QfbReduce","D0,G,p","QfbReduce","./ellQ2.gp.so"); GP;install("IndefiniteLLL","D0,G,DGDGp","IndefiniteLLL","./ellQ2.gp.so"); GP;install("IndefiniteLLL2","D0,G,DGDGp","IndefiniteLLL2","./ellQ2.gp.so"); GP;install("kermodp","D0,G,D0,G,","kermodp","./ellQ2.gp.so"); GP;install("Qfparam","D0,G,D0,G,DGp","Qfparam","./ellQ2.gp.so"); GP;install("LLLgoon3","D0,G,DGp","LLLgoon3","./ellQ2.gp.so"); GP;install("completebasis","D0,G,DG","completebasis","./ellQ2.gp.so"); GP;install("LLLgoon","D0,G,DGp","LLLgoon","./ellQ2.gp.so"); GP;install("QfWittinvariant","D0,G,D0,G,","QfWittinvariant","./ellQ2.gp.so"); GP;install("Qflisteinvariants","D0,G,DGp","Qflisteinvariants","./ellQ2.gp.so"); GP;install("Qfsolvemodp","D0,G,D0,G,p","Qfsolvemodp","./ellQ2.gp.so"); GP;install("Qfminim","D0,G,DGp","Qfminim","./ellQ2.gp.so"); GP;install("mymat","D0,G,","mymat","./ellQ2.gp.so"); GP;install("Qfbsqrtgauss","D0,G,D0,G,p","Qfbsqrtgauss","./ellQ2.gp.so"); GP;install("class2","D0,G,D0,G,D0,G,D0,G,p","class2","./ellQ2.gp.so"); GP;install("Qfsolve","D0,G,D0,G,p","Qfsolve","./ellQ2.gp.so"); GP;install("matdiagonalblock","D0,G,","matdiagonalblock","./ellQ2.gp.so"); GP;install("ellchangecurveinverse","D0,G,D0,G,","ellchangecurveinverse","./ellQ2.gp.so"); GP;install("ellchangepointinverse","D0,G,D0,G,","ellchangepointinverse","./ellQ2.gp.so"); GP;install("ellcomposeurst","D0,G,D0,G,","ellcomposeurst","./ellQ2.gp.so"); GP;install("ellinverturst","D0,G,","ellinverturst","./ellQ2.gp.so"); GP;install("mysubst","D0,G,D0,G,","mysubst","./ellQ2.gp.so"); GP;install("degre","D0,G,","degre","./ellQ2.gp.so"); GP;install("nfissquare","lD0,G,D0,G,p","nfissquare","./ellQ2.gp.so"); GP;install("nfsqrt","D0,G,D0,G,p","nfsqrt","./ellQ2.gp.so"); GP;install("sqrtrat","D0,G,","sqrtrat","./ellQ2.gp.so"); GP;install("polratroots","D0,G,","polratroots","./ellQ2.gp.so"); GP;addhelp(polratroots, "polratroots(pol): rational roots of the polynomial pol, according to the field of definition of its coefficients (Q, R, Qp, Fp)."); GP;install("ratpoint","D0,G,DGDGp","ratpoint","./ellQ2.gp.so"); GP;install("ratpoint2","D0,G,DGDGDGp","ratpoint2","./ellQ2.gp.so"); GP;install("listratpoint","D0,G,DGp","listratpoint","./ellQ2.gp.so"); GP;install("redquartic","D0,G,p","redquartic","./ellQ2.gp.so"); GP;addhelp(redquartic, "redquartic(pol): reduction of the quartic pol using Cremona-Stoll algorithm. Returns [p,M], where p is the reduced quartic and M is the GL2(Z) transformation. Also works with other degree polynomials."); GP;install("polrealrootsisolate","D0,G,","polrealrootsisolate","./ellQ2.gp.so"); GP;install("polrealrootsimprove","D0,G,D0,G,","polrealrootsimprove","./ellQ2.gp.so"); GP;addhelp(polrealrootsimprove, "ratpoint(pol,lim=1,singlepoint=1): search for rational points on y^2=pol(x), for about within the bounds given by lim. The coefficients of pol must be integral. If singlepoint=1, returns at most one point, otherwise as many as possible."); GP;install("polrootsmodpn","D0,G,D0,G,p","polrootsmodpn","./ellQ2.gp.so"); GP;install("ppinit","D0,G,D0,G,","ppinit","./ellQ2.gp.so"); GP;install("nfpsquareoddQ","lD0,G,D0,G,D0,G,p","nfpsquareoddQ","./ellQ2.gp.so"); GP;install("psquare","lD0,G,D0,G,p","psquare","./ellQ2.gp.so"); GP;install("lemma6","lD0,G,D0,G,D0,G,D0,G,p","lemma6","./ellQ2.gp.so"); GP;install("lemma7","lD0,G,D0,G,D0,G,p","lemma7","./ellQ2.gp.so"); GP;install("zpsoluble","lD0,G,D0,G,D0,G,D0,G,D0,G,D0,G,p","zpsoluble","./ellQ2.gp.so"); GP;install("qpsoluble","lD0,G,D0,G,p","qpsoluble","./ellQ2.gp.so"); GP;install("locallysoluble","lD0,G,p","locallysoluble","./ellQ2.gp.so"); GP;addhelp(locallysoluble, "locallysoluble(pol): returns 1 if y^2=pol(x) is everywhere locally soluble, 0 otherwise."); GP;install("LS2localimage","D0,G,D0,G,D0,G,p","LS2localimage","./ellQ2.gp.so"); GP;install("ellhalf","D0,G,D0,G,p","ellhalf","./ellQ2.gp.so"); GP;addhelp(ellhalf, "ellhalf(E,P): returns the vector of all points Q on the elliptic curve E such that 2Q = P"); GP;install("elltors2","D0,G,p","elltors2","./ellQ2.gp.so"); GP;addhelp(elltors2, "elltors2(E): for an elliptic curve E, returns the group E(K)[2], where K is the field of definition of the coefficients of E (Q, R, Qp or Fp)."); GP;install("elltorseven","D0,G,p","elltorseven","./ellQ2.gp.so"); GP;addhelp(elltorseven, "elltorseven(E): for an elliptic curve E, returns 2-Sylow subgroup of E(K)_tors, where K is the field of definition of the coefficients of E: (Q, R, Qp or Fp)."); GP;install("ellsort","D0,G,p","ellsort","./ellQ2.gp.so"); GP;addhelp(ellsort, "ellsort(v): v being a vector of points on some elliptic curve, returns the vector v sorted according to the naive height."); GP;install("ellremovetorsion","D0,G,D0,G,","ellremovetorsion","./ellQ2.gp.so"); GP;install("ellredgen","D0,G,D0,G,DGp","ellredgen","./ellQ2.gp.so"); GP;addhelp(ellredgen, "ellredgen(E,v): returns a vector of smallest possible points on the elliptic curve E generating the same subgroup as v, up to torsion."); GP;install("reducemodsquares","D0,G,D0,G,p","reducemodsquares","./ellQ2.gp.so"); GP;addhelp(reducemodsquares, "reducemodsquares(delta,d): delta being a t_POLMOD, returns another delta'=delta*z^2, such that delta' has a small coefficient in x^d."); GP;install("bnfpSelmer","D0,G,DGD0,G,p","bnfpSelmer","./ellQ2.gp.so"); GP;addhelp(bnfpSelmer, "bnfpSelmer(K,S,p): K being a number field given by bnfinit, S an ideal of K, and p a prime number, computes a set of generators of the group K(S,p) = { x in K^/K^p, v_P(x) = 0 (mod p) for all P coprime to S}"); GP;install("kersign","D0,G,D0,G,","kersign","./ellQ2.gp.so"); GP;install("kernorm","D0,G,D0,G,D0,G,","kernorm","./ellQ2.gp.so"); GP;install("elllocalimage","D0,G,D0,G,DGp","elllocalimage","./ellQ2.gp.so"); GP;install("ell2descent_gen","D0,G,D0,G,DGDGDGp","ell2descent_gen","./ellQ2.gp.so"); GP;addhelp(ell2descent_gen, "ell2descent_gen((E,bnf,k=1,help=[]): E is a vector of the form [0,A,0,B,C], (or the result of ellinit of such a vector) A,B,C integers such that x^3+A*x^2+B*x+C; bnf is the corresponding bnfinit(,1); Performs 2-descent on the elliptic curve Ek : k*y^2=x^3+A*x^2+B*x+C. See ?ellrank for the format of the output."); GP;install("afficheselmer","vD0,G,D0,G,D0,G,","afficheselmer","./ellQ2.gp.so"); GP;install("ellrankdebug","D0,G,D0,G,DGp","ellrankdebug","./ellQ2.gp.so"); GP;install("ellrank","D0,G,DGp","ellrank","./ellQ2.gp.so"); GP;addhelp(ellrank, "ellrank(E,help=[]): E is any elliptic curve defined over Q. Returns a vector [r,s,v], where r is a lower bound for the rank of E, s is the rank of its 2-Selmer group and v is a list of independant points in E(Q)/2E(Q). If help is a vector of nontrivial points on E, the result might be faster. This function might be used in conjunction with elltors2(E)"); GP;install("ell2descent_complete","D0,G,D0,G,D0,G,D0,G,p","ell2descent_complete","./ellQ2.gp.so"); GP;addhelp(ell2descent_complete, "ell2descent_complete(e1,e2,e3): Performs a complete 2-descent on the elliptic curve y^2 = (x-e1)*(x-e2)*(x-e3). See ?ellrank for the format of the output."); GP;install("ellcount","D0,G,D0,G,D0,G,DGp","ellcount","./ellQ2.gp.so"); GP;install("ell2descent_viaisog","D0,G,DGp","ell2descent_viaisog","./ellQ2.gp.so"); GP;addhelp(ell2descent_viaisog, "ell2descent_viaisog(E,help=[]): E is an elliptic curve of the form [0,a,0,b,0], with a, b integers. Performs a 2-descent via isogeny on E. See ?ellrank for the format of the output."); GP;install("nfsign_s","lD0,G,D0,G,D0,G,p","nfsign_s","./ellQ2.gp.so"); GP;install("nfpolratroots","D0,G,D0,G,","nfpolratroots","./ellQ2.gp.so"); GP;install("nfmodid2","D0,G,D0,G,D0,G,","nfmodid2","./ellQ2.gp.so"); GP;install("nfhilb2","D0,G,D0,G,D0,G,D0,G,p","nfhilb2","./ellQ2.gp.so"); GP;install("mynfhilbertp","D0,G,D0,G,D0,G,D0,G,p","mynfhilbertp","./ellQ2.gp.so"); GP;install("ideallistfactor","D0,G,D0,G,","ideallistfactor","./ellQ2.gp.so"); GP;install("mynfhilbert","lD0,G,D0,G,D0,G,p","mynfhilbert","./ellQ2.gp.so"); GP;install("initp","D0,G,D0,G,p","initp","./ellQ2.gp.so"); GP;install("deno","D0,G,","deno","./ellQ2.gp.so"); GP;install("nfratpoint","D0,G,D0,G,D0,G,DGp","nfratpoint","./ellQ2.gp.so"); GP;install("repres","D0,G,D0,G,p","repres","./ellQ2.gp.so"); GP;install("val","D0,G,D0,G,D0,G,","val","./ellQ2.gp.so"); GP;install("nfissquarep","D0,G,D0,G,D0,G,D0,G,p","nfissquarep","./ellQ2.gp.so"); GP;install("nfpsquareodd","lD0,G,D0,G,D0,G,p","nfpsquareodd","./ellQ2.gp.so"); GP;install("nfpsquare","lD0,G,D0,G,D0,G,D0,G,p","nfpsquare","./ellQ2.gp.so"); GP;install("nfpsquareq","lD0,G,D0,G,D0,G,D0,G,p","nfpsquareq","./ellQ2.gp.so"); GP;install("nflemma6","lD0,G,D0,G,D0,G,D0,G,D0,G,p","nflemma6","./ellQ2.gp.so"); GP;install("nflemma7","lD0,G,D0,G,D0,G,D0,G,D0,G,D0,G,p","nflemma7","./ellQ2.gp.so"); GP;install("nfzpsoluble","lD0,G,D0,G,D0,G,D0,G,D0,G,D0,G,p","nfzpsoluble","./ellQ2.gp.so"); GP;install("mynfeltmod","D0,G,D0,G,D0,G,","mynfeltmod","./ellQ2.gp.so"); GP;install("mynfeltreduce","D0,G,D0,G,D0,G,","mynfeltreduce","./ellQ2.gp.so"); GP;install("nfrandintmodid","D0,G,D0,G,","nfrandintmodid","./ellQ2.gp.so"); GP;install("nfrandint","D0,G,D0,G,","nfrandint","./ellQ2.gp.so"); GP;install("nfqpsolublebig","lD0,G,D0,G,D0,G,DGDGp","nfqpsolublebig","./ellQ2.gp.so"); GP;install("nfpolrootsmod","D0,G,D0,G,D0,G,","nfpolrootsmod","./ellQ2.gp.so"); GP;install("nfqpsoluble","lD0,G,D0,G,D0,G,p","nfqpsoluble","./ellQ2.gp.so"); GP;install("nflocallysoluble","lD0,G,D0,G,DGDGDGp","nflocallysoluble","./ellQ2.gp.so"); GP;install("nfellcount","D0,G,D0,G,D0,G,D0,G,D0,G,p","nfellcount","./ellQ2.gp.so"); GP;install("gettufu","D0,G,","gettufu","./ellQ2.gp.so"); GP;install("getfutu","D0,G,","getfutu","./ellQ2.gp.so"); GP;install("bnfell2descent_viaisog","D0,G,D0,G,p","bnfell2descent_viaisog","./ellQ2.gp.so"); GP;install("nfchinremain","D0,G,D0,G,D0,G,","nfchinremain","./ellQ2.gp.so"); GP;install("bnfqfsolve2","D0,G,D0,G,D0,G,DGp","bnfqfsolve2","./ellQ2.gp.so"); GP;install("bnfqfsolve","D0,G,D0,G,D0,G,D0,G,DGp","bnfqfsolve","./ellQ2.gp.so"); GP;install("bnfredquartique2","D0,G,D0,G,D0,G,D0,G,D0,G,","bnfredquartique2","./ellQ2.gp.so"); GP;install("bnfell2descent_gen","D0,G,D0,G,D0,G,DGDGDGDGp","bnfell2descent_gen","./ellQ2.gp.so"); GP;install("bnfellrank","D0,G,D0,G,DGDGDGp","bnfellrank","./ellQ2.gp.so"); GP;install("bnfell2descent_complete","D0,G,D0,G,D0,G,D0,G,DGDGp","bnfell2descent_complete","./ellQ2.gp.so"); */ void init_ellQ2(void); GEN QfbReduce(GEN M, long prec); GEN IndefiniteLLL(GEN G, GEN c, GEN base, long prec); GEN IndefiniteLLL2(GEN G, GEN c, GEN base, long prec); GEN kermodp(GEN M, GEN p); GEN Qfparam(GEN G, GEN sol, GEN fl, long prec); GEN LLLgoon3(GEN G, GEN c, long prec); GEN completebasis(GEN v, GEN redflag); GEN LLLgoon(GEN G, GEN c, long prec); GEN QfWittinvariant(GEN G, GEN p); GEN Qflisteinvariants(GEN G, GEN fa, long prec); GEN Qfsolvemodp(GEN G, GEN p, long prec); GEN Qfminim(GEN G, GEN factdetG, long prec); GEN mymat(GEN qfb); GEN Qfbsqrtgauss(GEN G, GEN factdetG, long prec); GEN class2(GEN D, GEN factdetG, GEN Winvariants, GEN U2, long prec); GEN Qfsolve(GEN G, GEN factD, long prec); GEN matdiagonalblock(GEN v); GEN ellchangecurveinverse(GEN ell, GEN v); GEN ellchangepointinverse(GEN pt, GEN v); GEN ellcomposeurst(GEN urst1, GEN urst2); GEN ellinverturst(GEN urst); GEN mysubst(GEN polsu, GEN subsx); GEN degre(GEN idegre); long nfissquare(GEN nf, GEN a, long prec); GEN nfsqrt(GEN nf, GEN a, long prec); GEN sqrtrat(GEN a); GEN polratroots(GEN pol); GEN ratpoint(GEN pol, GEN lim, GEN singlepoint, long prec); GEN ratpoint2(GEN pol, GEN lim, GEN singlepoint, GEN redflag, long prec); GEN listratpoint(GEN pol, GEN redflag, long prec); GEN redquartic(GEN pol, long prec); GEN polrealrootsisolate(GEN pol); GEN polrealrootsimprove(GEN pol, GEN v); GEN polrootsmodpn(GEN pol, GEN p, long prec); GEN ppinit(GEN nf, GEN p); long nfpsquareoddQ(GEN nf, GEN a, GEN pr, long prec); long psquare(GEN a, GEN p, long prec); long lemma6(GEN pol, GEN p, GEN nu, GEN xx, long prec); long lemma7(GEN pol, GEN nu, GEN xx, long prec); long zpsoluble(GEN pol, GEN p, GEN nu, GEN pnu, GEN x0, GEN pnup, long prec); long qpsoluble(GEN pol, GEN p, long prec); long locallysoluble(GEN pol, long prec); GEN LS2localimage(GEN nf, GEN gen, GEN pp, long prec); GEN ellhalf(GEN ell, GEN P, long prec); GEN elltors2(GEN ell, long prec); GEN elltorseven(GEN ell, long prec); GEN ellsort(GEN listpts, long prec); GEN ellremovetorsion(GEN ell, GEN listgen); GEN ellredgen(GEN ell0, GEN listgen, GEN K, long prec); GEN reducemodsquares(GEN delta, GEN d, long prec); GEN bnfpSelmer(GEN bnf, GEN S, GEN p, long prec); GEN kersign(GEN gen, GEN rootapprox); GEN kernorm(GEN gen, GEN S, GEN p); GEN elllocalimage(GEN nf, GEN pp, GEN K, long prec); GEN ell2descent_gen(GEN ell, GEN bnf, GEN K, GEN help, GEN redflag, long prec); void afficheselmer(GEN m1, GEN m2, GEN tors2); GEN ellrankdebug(GEN ell, GEN lev, GEN help, long prec); GEN ellrank(GEN ell, GEN help, long prec); GEN ell2descent_complete(GEN e1, GEN e2, GEN e3, GEN help, long prec); GEN ellcount(GEN c, GEN d, GEN KS2gen, GEN listpointstriv, long prec); GEN ell2descent_viaisog(GEN ell, GEN help, long prec); long nfsign_s(GEN nf, GEN a, GEN i, long prec); GEN nfpolratroots(GEN nf, GEN pol); GEN nfmodid2(GEN nf, GEN a, GEN ideal); GEN nfhilb2(GEN nf, GEN a, GEN b, GEN p, long prec); GEN mynfhilbertp(GEN nf, GEN a, GEN b, GEN p, long prec); GEN ideallistfactor(GEN nf, GEN listfact); long mynfhilbert(GEN nf, GEN a, GEN b, long prec); GEN initp(GEN nf, GEN p, long prec); GEN deno(GEN num); GEN nfratpoint(GEN nf, GEN pol, GEN lim, GEN singlepoint, long prec); GEN repres(GEN nf, GEN p, long prec); GEN val(GEN nf, GEN num, GEN p); GEN nfissquarep(GEN nf, GEN a, GEN p, GEN q, long prec); long nfpsquareodd(GEN nf, GEN a, GEN p, long prec); long nfpsquare(GEN nf, GEN a, GEN p, GEN zinit, long prec); long nfpsquareq(GEN nf, GEN a, GEN p, GEN q, long prec); long nflemma6(GEN nf, GEN pol, GEN p, GEN nu, GEN xx, long prec); long nflemma7(GEN nf, GEN pol, GEN p, GEN nu, GEN xx, GEN zinit, long prec); long nfzpsoluble(GEN nf, GEN pol, GEN p, GEN nu, GEN pnu, GEN x0, long prec); GEN mynfeltmod(GEN nf, GEN a, GEN b); GEN mynfeltreduce(GEN nf, GEN a, GEN id); GEN nfrandintmodid(GEN nf, GEN id); GEN nfrandint(GEN nf, GEN borne); long nfqpsolublebig(GEN nf, GEN pol, GEN p, GEN ap, GEN b, long prec); GEN nfpolrootsmod(GEN nf, GEN pol, GEN p); long nfqpsoluble(GEN nf, GEN pol, GEN p, long prec); long nflocallysoluble(GEN nf, GEN pol, GEN r, GEN a, GEN b, long prec); GEN nfellcount(GEN nf, GEN c, GEN d, GEN KS2gen, GEN pointstriv, long prec); GEN gettufu(GEN bnf); GEN getfutu(GEN bnf); GEN bnfell2descent_viaisog(GEN bnf, GEN ell, long prec); GEN nfchinremain(GEN nf, GEN b, GEN fact); GEN bnfqfsolve2(GEN bnf, GEN aleg, GEN bleg, GEN auto_s, long prec); GEN bnfqfsolve(GEN bnf, GEN aleg, GEN bleg, GEN flag3, GEN auto_s, long prec); GEN bnfredquartique2(GEN bnf, GEN pol, GEN r, GEN a, GEN b); GEN bnfell2descent_gen(GEN bnf, GEN ell, GEN ext, GEN help, GEN bigflag, GEN flag3, GEN auto_s, long prec); GEN bnfellrank(GEN bnf, GEN ell, GEN help, GEN bigflag, GEN flag3, long prec); GEN bnfell2descent_complete(GEN bnf, GEN e1, GEN e2, GEN e3, GEN flag3, GEN auto_s, long prec); /*End of prototype*/ static GEN DEBUGLEVEL_qfsolve; static GEN DEBUGLEVEL_ell; static GEN LIM1; static GEN LIM3; static GEN LIMTRIV; static GEN COMPLETE; static GEN MAXPROB; static GEN LIMBIGPRIME; static GEN ELLREDGENFLAG; static GEN BIGINT; static GEN NBIDEAUX; /*End of global vars*/ void init_ellQ2(void) /* void */ { pari_sp ltop = avma; DEBUGLEVEL_qfsolve = pol_x(fetch_user_var("DEBUGLEVEL_qfsolve")); DEBUGLEVEL_ell = pol_x(fetch_user_var("DEBUGLEVEL_ell")); LIM1 = pol_x(fetch_user_var("LIM1")); LIM3 = pol_x(fetch_user_var("LIM3")); LIMTRIV = pol_x(fetch_user_var("LIMTRIV")); COMPLETE = pol_x(fetch_user_var("COMPLETE")); MAXPROB = pol_x(fetch_user_var("MAXPROB")); LIMBIGPRIME = pol_x(fetch_user_var("LIMBIGPRIME")); ELLREDGENFLAG = pol_x(fetch_user_var("ELLREDGENFLAG")); BIGINT = pol_x(fetch_user_var("BIGINT")); NBIDEAUX = pol_x(fetch_user_var("NBIDEAUX")); /*\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ */ /* Copyright (C) 2011 Denis Simon */ /* */ /* Distributed under the terms of the GNU General Public License (GPL) */ /* */ /* This code is distributed in the hope that it will be useful, */ /* but WITHOUT ANY WARRANTY; without even the implied warranty of */ /* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU */ /* General Public License for more details. */ /* */ /* The full text of the GPL is available at: */ /* */ /* http://www.gnu.org/licenses/ */ /*\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ */ /* Auteur : Denis SIMON -> simon@math.unicaen.fr adresse du fichier : www.math.unicaen.fr/~simon/ellQ.gp ********************************************* * VERSION 24/01/2011 * ********************************************* \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ \\ English \\ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ This package provides functions to compute the rank of elliptic curves over Q using 2-descent. This package requires the other package qfsolve.gp downloadable at www.math.unicaen.fr/~simon/qfsolve.gp They can be run under GP by the commands gp > \r qfsolve.gp gp > \r ellQ.gp The main function is ellrank(), which takes as an argument any elliptic curve in the form [a1,a2,a3,a4,a6] the result is a vector [r,s,v], where r is a lower bound for the rank, s is the rank of the 2-Selmer group v is a set of independant points in E(Q)/2E(Q). Example: gp > ell = [1,2,3,4,5]; gp > ellrank(ell) %1 = [1, 1, [[1,2]] In this example, the rank is exactly 1, and [1,2] has infinite order. more details on the computations may be obtained by setting DEBUGLEVEL_ell = 1 (the higher value, the more details) Other functions: ell2descent_complete, ell2descent_gen, ell2descent_viaisog, ellhalf, ellredgen, ellsort, elltors2, elltorseven, locallysoluble, polratroots, ratpoint, redquartic, bnfpSelmer, reducemodsquares Quick information is obtained by typing gp > ?NameOfTheFunction \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ \\ Comment utiliser ce programme ? \\ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Programme de calcul du rang des courbes elliptiques sur Q. langage : GP pour l'utiliser, lancer gp, puis taper \r ellQ.gp Ce programme utilise le module de resolution des formes quadratiques situe a l'adresse www.math.unicaen.fr/~simon/qfsolve.gp Il faut donc aussi taper : \r qfsolve.gp \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ \\ Description des principales fonctions \\ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ Explications succintes : La fonction ellrank() accepte toutes les courbes sous la forme [a1,a2,a3,a4,a6] Les coefficients peuvent etre entiers ou non. L'algorithme utilise est celui de la 2-descente. La 2-torsion peut etre quelconque. Il suffit de taper : gp > ell = [a1,a2,a3,a4,a6]; gp > ellrank(ell) Retourne un vecteur [r,s,v] ou r est le rang probable (c'est toujours une minoration du rang), s est le 2-rang du groupe de Selmer, v est une liste de points independants dans E(Q)/2E(Q). Exemple : gp > ell = [1,2,3,4,5]; gp > ellrank(ell) %1 = [1, 1, [[1,2]] Ici, le rang est exactement 1, et le point [1,2] est d'ordre infini. Courbes de la forme : k*y^2 = x^3+A*x^2+B*x+C sans 2-torsion, A,B,C entiers. gp > bnf = bnfinit(x^3+A*x^2+B*x+C); gp > ell = ellinit([0,A,0,B,C],1); gp > rank = ell2descent_gen(ell,bnf,k); Courbes avec #E[2](Q) >= 2 : ell doit etre sous la forme y^2 = x^3 + A*^2 + B*x avec A et B entiers. gp > ell = [0,A,0,B,0] gp > ell2descent_viaisog(ell) = algorithme de la 2-descente par isogenies Attention A et B doivent etre entiers Courbes avec #E[2](Q) = 4 : y^2 = (x-e1)*(x-e2)*(x-e3) gp > ell2descent_complete(e1,e2,e3) = algorithme de la 2-descente complete Attention : les ei doivent etre entiers. \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ \\ Autres fonctions utiles \\ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ elltors2(E) : determine le groupe E[2](Q) elltorseven(E) : determine le groupe E[2^*](Q) ellhalf(E,P) : liste les points Q tels que 2Q = P ellredgen(E,v) : reduction des points de v sur E locallysoluble(pol): teste si y^2=pol(x) est ELS ratpoint(pol,lim): cherche un point sur y^2=pol(x) redquartic(pol): reduction de la quartique pol polratroots(pol) : liste les solutions rationnelles de pol Aide en ligne : ?NomDeLaFonction \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ \\ Affichage des calculs \\ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ On peut avoir plus ou moins de details de calculs avec DEBUGLEVEL_ell = 0; DEBUGLEVEL_ell = 1; 2; 3;... */ /*\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ */ /* Copyright (C) 2011 Denis Simon */ /* */ /* Distributed under the terms of the GNU General Public License (GPL) */ /* */ /* This code is distributed in the hope that it will be useful, */ /* but WITHOUT ANY WARRANTY; without even the implied warranty of */ /* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU */ /* General Public License for more details. */ /* */ /* The full text of the GPL is available at: */ /* */ /* http://www.gnu.org/licenses/ */ /*\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ */ /* Author: Denis SIMON -> simon@math.unicaen.fr address of the file: www.math.unicaen.fr/~simon/qfsolve.gp ********************************************* * VERSION 21/02/2011 * ********************************************* \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ \\ English help \\ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ This package provides functions to solve quadratic equations over Q. language: GP It can be run under GP by the command gp > \r qfsolve.gp This package contains 4 main functions: - Qfsolve(G,factD): solve over Q the quadratic equation X~*G*X = 0. G must be a symmetric matrix n*n, with coefficients in Z. If no solution exists, the output is a prime number indicating that there is no solution in the local field Q_p (-1 for the reals). This algorithm requires the factorization of -abs(2*matdet(G)). If this factorization is known, one can give it as factD (the second argument of the function) and save a lot of time. - Qfparam(G,sol,fl): parametrization by quadratic forms of the solutions of the ternary quadratic form G, using the particular solution sol. If fl>0, then the 'fl'th form is reduced. - IndefiniteLLL(G,c): Solve or reduce the quadratic form G with integral coefficients. G might be definite or indefinite. This is an LLL-type algorithm with a constant 1/40, la 'fl'eme forme quadratique est reduite. - IndefiniteLLL(G,c): pour resoudre ou reduire la forme quadratique G a coefficients entiers. Il s'agit d'un algorithme type LLL, avec la constante 1/4= 0) pari_printf("ellchangecurveinverse\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("ellchangepointinverse\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("ellcomposeurst\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("ellinverturst\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("mysubst\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("degre\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("nfissquare\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("nfsqrt\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("sqrtrat\n"); /* \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ */ /* FUNCTIONS SPECIFIC TO ellQ.gp \\ */ /* \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ */ /* \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ */ /* FUNCTIONS FOR POLYNOMIALS \\ */ /* \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ */ if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("polratroots\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("ratpoint\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("ratpoint2\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("listratpoint\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("redquartic\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("polrealrootsisolate\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("polrealrootsimprove\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("polrootsmodpn\n"); /* \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ */ /* FUNCTIONS FOR LOCAL COMPUTATIONS \\ */ /* \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ */ if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("ppinit\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("nfpsquareoddQ\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("psquare\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("lemma6\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("lemma7\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("zpsoluble\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("qpsoluble\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("locallysoluble\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("LS2localimage\n"); /* \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ */ /* GENERIC FUNCTIONS FOR ELLIPTIC CURVES \\ */ /* \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ */ if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("ellhalf\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("elltors2\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("elltorseven\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("ellsort\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("ellremovetorsion\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("ellredgen\n"); /* \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ */ /* FUNCTIONS FOR NUMBER FIELDS \\ */ /* \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ */ if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("reducemodsquares\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("bnfpSelmer\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("kersign\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("kernorm\n"); /* \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ */ /* FUNCTIONS FOR 2-DESCENT \\ */ /* \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ */ if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("elllocalimage\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("ell2descent_gen\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("afficheselmer\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("ellrank\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("ell2descent_complete\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("ellcount\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("ell2descent_viaisog\n"); /* \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ */ /* HELP MESSAGES \\ */ /* \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ */ /* functions for elliptic curves */ /* functions for polynomials */ /* functions for number fields */ /* others */ /*\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ */ /* Copyright (C) 2007 Denis Simon */ /* */ /* Distributed under the terms of the GNU General Public License (GPL) */ /* */ /* This code is distributed in the hope that it will be useful, */ /* but WITHOUT ANY WARRANTY; without even the implied warranty of */ /* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU */ /* General Public License for more details. */ /* */ /* The full text of the GPL is available at: */ /* */ /* http://www.gnu.org/licenses/ */ /*\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ */ /* */ /* Auteur: */ /* Denis SIMON -> simon@math.unicaen.fr */ /* adresse du fichier: */ /* www.math.unicaen.fr/~simon/ell.gp */ /* */ /* ********************************************* */ /* * VERSION 25/03/2009 * */ /* ********************************************* */ /* */ /* Programme de calcul du rang des courbes elliptiques */ /* dans les corps de nombres. */ /* langage: GP */ /* pour l'utiliser, lancer gp, puis taper */ /* \r ell.gp */ /* */ /* */ /* Explications succintes : */ /* definition du corps : */ /* bnf=bnfinit(y^2+1); */ /* (il est indispensable que la variable soit y). */ /* on peut ensuite poser : */ /* X = Mod(y,bnf.pol); */ /* */ /* La fonction bnfellrank() accepte toutes les courbes sous la forme */ /* [a1,a2,a3,a4,a6] */ /* Les coefficients peuvent etre entiers ou non. */ /* L'algorithme utilise est celui de la 2-descente. */ /* La 2-torsion peut etre quelconque. */ /* Il suffit de taper : */ /* */ /* gp > ell = [a1,a2,a3,a4,a6]; */ /* gp > bnfellrank(bnf,ell) */ /* */ /* Retourne un vecteur [r,s,vec] */ /* ou r est le rang probable (c'est toujours une minoration du rang), */ /* s est le 2-rang du groupe de Selmer, */ /* vec est une liste de points dans E(K)/2E(K). */ /* */ /* Courbes avec #E[2](K) >= 2: */ /* ell doit etre sous la forme */ /* y^2 = x^3 + A*^2 + B*x */ /* avec A et B entiers algebriques */ /* gp > ell = [0,A,0,B,0] */ /* gp > bnfell2descent_viaisog(ell) */ /* = algorithme de la 2-descente par isogenies */ /* Attention A et B doivent etre entiers */ /* */ /* Courbes avec #E[2](K) = 4: y^2 = (x-e1)*(x-e2)*(x-e3) */ /* -> bnfell2descent_complete(bnf,e1,e2,e3); */ /* = algorithme de la 2-descente complete */ /* Attention: les ei doivent etre entiers algebriques. */ /* */ /* */ /* On peut avoir plus ou moins de details de calculs avec */ /* DEBUGLEVEL_ell = 0; */ /* DEBUGLEVEL_ell = 1; 2; 3;... */ /* */ /* */ /* Variables globales usuelles */ /* */ DEBUGLEVEL_ell = gen_1; /* pour avoir plus ou moins de details */ LIM1 = gen_2; /* limite des points triviaux sur les quartiques */ LIM3 = stoi(4); /* limite des points sur les quartiques ELS */ LIMTRIV = gen_2; /* limite des points triviaux sur la courbe elliptique */ /* */ /* Variables globales techniques */ /* */ BIGINT = stoi(32000); /* l'infini */ MAXPROB = stoi(20); LIMBIGPRIME = stoi(30); /* pour distinguer un petit nombre premier d'un grand */ /* utilise un test probabiliste pour les grands */ /* si LIMBIGPRIME = 0, n'utilise aucun test probabiliste */ NBIDEAUX = stoi(10); /* */ /* Programmes */ /* */ /* */ /* Fonctions communes ell.gp et ellQ.gp */ /* */ /* { ellinverturst(urst) = local(u = urst[1], r = urst[2], s = urst[3], t = urst[4]); [1/u,-r/u^2,-s/u,(r*s-t)/u^3]; } */ /* { ellchangecurveinverse(ell,v) = ellchangecurve(ell,ellinverturst(v)); } { ellchangepointinverse(pt,v) = ellchangepoint(pt,ellinverturst(v)); } */ /* { ellcomposeurst(urst1,urst2) = local(u1 = urst1[1], r1 = urst1[2], s1 = urst1[3], t1 = urst1[4], u2 = urst2[1], r2 = urst2[2], s2 = urst2[3], t2 = urst2[4]); [u1*u2,u1^2*r2+r1,u1*s2+s1,u1^3*t2+s1*u1^2*r2+t1]; } */ /* if( DEBUGLEVEL_ell >= 4, print("mysubst")); { mysubst(polsu,subsx) = if( type(lift(polsu)) == "t_POL", return(simplify(subst(lift(polsu),variable(lift(polsu)),subsx)) ) , return(simplify(lift(polsu)))); } */ /* MODI nfsign is a GP function, must modify */ if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("nfsign_s\n"); /* if( DEBUGLEVEL_ell >= 4, print("degre")); { degre(idegre) = local(ideg = idegre, jdeg = 0); while( ideg >>= 1, jdeg++); return(jdeg); } */ /* if( DEBUGLEVEL_ell >= 4, print("nfissquare")); { nfissquare(nf, a) = #nfsqrt(nf,a) > 0; } if( DEBUGLEVEL_ell >= 4, print("nfsqrt")); { nfsqrt( nf, a) = \\ si a est un carre, renvoie [sqrt(a)], sinon []. local(alift,ta,res,pfact); if( DEBUGLEVEL_ell >= 5, print("entree dans nfsqrt ",a)); if( a==0 || a==1, if( DEBUGLEVEL_ell >= 5, print("fin de nfsqrt")); return([a])); alift = lift(a); ta = type(a); if( !poldegree(alift), alift = polcoeff(alift,0)); if( type(alift) != "t_POL", if( issquare(alift), if( DEBUGLEVEL_ell >= 5, print("fin de nfsqrt")); return([sqrtrat(alift)]))); if( poldegree(nf.pol) <= 1, if( DEBUGLEVEL_ell >= 5, print("fin de nfsqrt")); return([])); if( ta == "t_POL", a = Mod(a,nf.pol)); \\ tous les plgements reels doivent etre >0 for( i = 1, nf.r1, if( nfsign_s(nf,a,i) < 0, if( DEBUGLEVEL_ell >= 5, print("fin de nfsqrt")); return([]))); \\ factorisation sur K du polynome X^2-a : if( variable(nf.pol) == x, py = subst(nf.pol,x,y); pfact = lift(factornf(x^2-mysubst(alift,Mod(y,py)),py)[1,1]) , pfact = lift(factornf(x^2-a,nf.pol)[1,1])); if( poldegree(pfact) == 2, if( DEBUGLEVEL_ell >= 5, print("fin de nfsqrt")); return([])); if( DEBUGLEVEL_ell >= 5, print("fin de nfsqrt")); return([subst(polcoeff(pfact,0),y,Mod(variable(nf.pol),nf.pol))]); } */ /* if( DEBUGLEVEL_ell >= 4, print("sqrtrat")); { sqrtrat(a) = sqrtint(numerator(a))/sqrtint(denominator(a)); } */ /* */ /* Fonctions propres a ell.gp */ /* */ if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("nfpolratroots\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("nfmodid2\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("nfhilb2\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("mynfhilbertp\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("ideallistfactor\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("mynfhilbert\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("initp\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("deno\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("nfratpoint\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("repres\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("val\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("nfissquarep\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("nfpsquareodd\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("nfpsquare\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("nfpsquareq\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("nflemma6\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("nflemma7\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("nfzpsoluble\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("mynfeltmod\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("mynfeltreduce\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("nfrandintmodid\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("nfrandint\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("nfqpsolublebig\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("nfpolrootsmod\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("nfqpsoluble\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("nflocallysoluble\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("nfellcount\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("bnfell2descent_viaisog\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("nfchinremain\n"); /* MODI must change auto to auto_s: reserved C */ if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("bnfqfsolve2\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("bnfqfsolve\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("bnfredquartique2\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("bnfell2descent_gen\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("bnfellrank\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("bnfell2descent_complete\n"); gerepileall(ltop, 11, &DEBUGLEVEL_qfsolve, &DEBUGLEVEL_ell, &LIM1, &LIM3, &LIMTRIV, &COMPLETE, &MAXPROB, &LIMBIGPRIME, &ELLREDGENFLAG, &BIGINT, &NBIDEAUX); return; } /* \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ */ /* SCRIPT \\ */ /* \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ */ GEN QfbReduce(GEN M, long prec) { pari_sp ltop = avma; GEN a = gen_0, b = gen_0, c = gen_0, H = gen_0, test = gen_0, di = gen_0, q = gen_0, r = gen_0, nexta = gen_0, nextb = gen_0, nextc = gen_0, aux = gen_0; if (gcmpgs(DEBUGLEVEL_qfsolve, 5) >= 0) pari_printf(" starting QfbReduce with %Ps\n", M); a = gcopy(gcoeff(M, 1, 1)); b = gcopy(gcoeff(M, 1, 2)); c = gcopy(gcoeff(M, 2, 2)); H = matid(2); test = gen_1; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); while (!gequal0(test) && !gequal0(a)) { di = divrem(b, a, -1); q = gcopy(gel(di, 1)); r = gcopy(gel(di, 2)); if (gcmp(gmulsg(2, r), gabs(a, prec)) > 0) { r = gsub(r, gabs(a, prec)); q = gaddgs(q, gsigne(a)); } gel(H, 2) = gsub(gel(H, 2), gmul(q, gel(H, 1))); nextc = a; nextb = gneg(r); nexta = gadd(gmul(gsub(nextb, b), q), c); if (!gequal0(test = stoi(gcmp(gabs(nexta, prec), gabs(a, prec)) < 0))) { c = nextc; b = nextb; a = nexta; aux = gcopy(gel(H, 1)); gel(H, 1) = gneg(gel(H, 2)); gel(H, 2) = gcopy(aux); } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 12, &di, &q, &r, &H, &nextc, &nextb, &nexta, &test, &c, &b, &a, &aux); } } if (gcmpgs(DEBUGLEVEL_qfsolve, 5) >= 0) pari_printf(" end of QfbReduce with %Ps\n", H); H = gerepilecopy(ltop, H); return H; } GEN IndefiniteLLL(GEN G, GEN c, GEN base, long prec) { pari_sp ltop = avma; GEN n = gen_0, M = gen_0, QD = gen_0, M1 = gen_0, S = gen_0, red = gen_0, p1 = gen_0, p2 = gen_0; GEN p3 = gen_0, p4 = gen_0; /* vec */ if (!c) c = gen_1; if (!base) base = gen_0; n = stoi(glength(G)); M = matid(gtos(n)); QD = gcopy(G); p1 = gsubgs(n, 1); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN i = gen_0, p5 = gen_0; long l6; /* lg */ long l7; GEN p8 = gen_0; /* vec */ for (i = gen_1; gcmp(i, p1) <= 0; i = gaddgs(i, 1)) { if (gequal0(gcoeff(QD, gtos(i), gtos(i)))) { p5 = IndefiniteLLL2(G, c, base, prec); p5 = gerepilecopy(ltop, p5); return p5; } M1 = matid(gtos(n)); p8 = gdiv(gneg(rowcopy(QD, gtos(i))), gcoeff(QD, gtos(i), gtos(i))); l6 = lg(M1); for (l7 = 1; l7 < l6; ++l7) gcoeff(M1, gtos(i), l7) = gcopy(gel(p8, l7)); gcoeff(M1, gtos(i), gtos(i)) = gen_1; M = gmul(M, M1); QD = gmul(gmul(gtrans(M1), QD), M1); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 6, &i, &p5, &M1, &p8, &M, &QD); } } M = ginv(M); QD = gmul(gmul(gtrans(M), gabs(QD, prec)), M); S = qflllgram0(gdiv(QD, content(QD)), 0); red = IndefiniteLLL2(gmul(gmul(gtrans(S), G), S), c, base, prec); if (typ(red) == t_COL) { p2 = gmul(S, red); p2 = gerepilecopy(ltop, p2); return p2; } if (glength(red) == 3) { p3 = cgetg(4, t_VEC); gel(p3, 1) = gcopy(gel(red, 1)); gel(p3, 2) = gmul(S, gel(red, 2)); gel(p3, 3) = gmul(S, gel(red, 3)); p3 = gerepilecopy(ltop, p3); return p3; } p4 = cgetg(3, t_VEC); gel(p4, 1) = gcopy(gel(red, 1)); gel(p4, 2) = gmul(S, gel(red, 2)); p4 = gerepilecopy(ltop, p4); return p4; } GEN IndefiniteLLL2(GEN G, GEN c, GEN base, long prec) { pari_sp ltop = avma; GEN n = gen_0, H = gen_0, M = gen_0, A = gen_0, aux = gen_0, sol = gen_0, k = gen_0, nextk = gen_0, swap = gen_0, q = gen_0, di = gen_0, HM = gen_0, aux1 = gen_0, aux2 = gen_0, Mkk1 = gen_0, bk1new = gen_0, Mkk1new = gen_0, newG = gen_0; GEN p1 = gen_0, p2 = gen_0, p3 = gen_0; /* vec */ if (!c) c = gen_1; if (!base) base = gen_0; n = stoi(glength(G)); if (gcmpgs(DEBUGLEVEL_qfsolve, 3) >= 0) pari_printf(" LLL dim %Ps avec |G| = %Ps\n", n, gdiv(glog(vecmax(gabs(G, prec)), prec), glog(stoi(10), prec))); if (gcmpgs(DEBUGLEVEL_qfsolve, 4) >= 0) { pari_printf(" LLL with \n"); pari_printf("%Ps\n", G); } /* MODI2.1 */ if (gcmpgs(n, 1) <= 0) { p1 = cgetg(3, t_VEC); gel(p1, 1) = gcopy(G); gel(p1, 2) = matid(gtos(n)); p1 = gerepilecopy(ltop, p1); return p1; } H = gcopy(M = matid(gtos(n))); { long l4, l5; p2 = cgetg(gtos(n)+1, t_MAT); for (l5 = 1; gcmpsg(l5, n) <= 0; ++l5) { gel(p2, l5) = cgetg(gtos(n)+1, t_COL); for (l4 = 1; gcmpsg(l4, n) <= 0; ++l4) gcoeff(p2, l4, l5) = gen_0; } } A = p2; /* compute Gram-Schmidt */ { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN i = gen_0; GEN p6 = gen_0; /* vec */ GEN p7 = gen_0, p8 = gen_0; for (i = gen_1; gcmp(i, n) <= 0; i = gaddgs(i, 1)) { if (gequal0(gcoeff(A, gtos(i), gtos(i)) = gcopy(gcoeff(G, gtos(i), gtos(i))))) { if (!gequal0(base)) { aux = gcopy(gel(H, 1)); gel(H, 1) = gcopy(gel(H, gtos(i))); gel(H, gtos(i)) = gneg(aux); p6 = cgetg(4, t_VEC); gel(p6, 1) = gmul(gmul(gtrans(H), G), H); gel(p6, 2) = gcopy(H); gel(p6, 3) = gcopy(gel(H, 1)); p6 = gerepilecopy(ltop, p6); return p6; } else { p7 = gcopy(gel(M, gtos(i))); p7 = gerepilecopy(ltop, p7); return p7; } } p8 = gsubgs(i, 1); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN j = gen_0, p9 = gen_0, p10 = gen_0; GEN p11 = gen_0; /* vec */ for (j = gen_1; gcmp(j, p8) <= 0; j = gaddgs(j, 1)) { p9 = gsubgs(j, 1); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN k = gen_0; p10 = gen_0; for (k = gen_1; gcmp(k, p9) <= 0; k = gaddgs(k, 1)) { p10 = gadd(p10, gmul(gcoeff(M, gtos(j), gtos(k)), gcoeff(A, gtos(i), gtos(k)))); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &p10, &k); } } gcoeff(A, gtos(i), gtos(j)) = gsub(gcoeff(G, gtos(i), gtos(j)), p10); gcoeff(M, gtos(i), gtos(j)) = gdiv(gcoeff(A, gtos(i), gtos(j)), gcoeff(A, gtos(j), gtos(j))); gcoeff(A, gtos(i), gtos(i)) = gsub(gcoeff(A, gtos(i), gtos(i)), gmul(gcoeff(M, gtos(i), gtos(j)), gcoeff(A, gtos(i), gtos(j)))); if (gequal0(gcoeff(A, gtos(i), gtos(i)))) { sol = gcopy(gel(gtrans(ginv(M)), gtos(i))); sol = gdiv(sol, content(sol)); if (!gequal0(base)) { H = completebasis(sol, NULL); aux = gcopy(gel(H, 1)); gel(H, 1) = gcopy(gel(H, gtos(n))); gel(H, gtos(n)) = gneg(aux); p11 = cgetg(4, t_VEC); gel(p11, 1) = gmul(gmul(gtrans(H), G), H); gel(p11, 2) = gcopy(H); gel(p11, 3) = gcopy(gel(H, 1)); p11 = gerepilecopy(ltop, p11); return p11; } else { sol = gerepilecopy(ltop, sol); return sol; } } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 9, &j, &p9, &p10, &A, &M, &sol, &H, &aux, &p11); } } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 9, &i, &A, &aux, &H, &p6, &p7, &p8, &M, &sol); } } /* LLL loop */ k = gen_2; nextk = gen_1; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN p12 = gen_0; while (gcmp(k, n) <= 0) { swap = gen_1; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN p13 = gen_0; long l14; GEN p15 = gen_0; /* vec */ GEN p16 = gen_0, p17 = gen_0, p18 = gen_0; GEN p19 = gen_0; /* vec */ while (!gequal0(swap)) { swap = gen_0; /* red(k,k-1); */ if (!gequal0(q = ground(gcoeff(M, gtos(k), gtos(gsubgs(k, 1)))))) { p13 = gsubgs(k, 2); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN i = gen_0; long l20; for (i = gen_1; gcmp(i, p13) <= 0; i = gaddgs(i, 1)) { l20 = gtos(gsubgs(k, 1)); gcoeff(M, gtos(k), gtos(i)) = gsub(gcoeff(M, gtos(k), gtos(i)), gmul(q, gcoeff(M, l20, gtos(i)))); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &i, &M); } } l14 = gtos(gsubgs(k, 1)); gcoeff(M, gtos(k), l14) = gsub(gcoeff(M, gtos(k), l14), q); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN i = gen_0; long l21, l22; for (i = gen_1; gcmp(i, n) <= 0; i = gaddgs(i, 1)) { l21 = gtos(gsubgs(k, 1)); gcoeff(A, gtos(k), gtos(i)) = gsub(gcoeff(A, gtos(k), gtos(i)), gmul(q, gcoeff(A, l21, gtos(i)))); l22 = gtos(gsubgs(k, 1)); gcoeff(H, gtos(i), gtos(k)) = gsub(gcoeff(H, gtos(i), gtos(k)), gmul(q, gcoeff(H, gtos(i), l22))); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 3, &i, &A, &H); } } } /* preparation of swap(k,k-1) */ if (!gequal0(gissquare(di = gmul(gneg(gcoeff(A, gtos(gsubgs(k, 1)), gtos(gsubgs(k, 1)))), gcoeff(A, gtos(k), gtos(k)))))) { /* di is the determinant of matr */ /* We find a solution */ HM = gtrans(ginv(M)); aux1 = sqrtint(numer(di)); aux2 = sqrtint(denom(di)); sol = gadd(gmul(aux1, gel(HM, gtos(gsubgs(k, 1)))), gmul(gmul(aux2, gcoeff(A, gtos(gsubgs(k, 1)), gtos(gsubgs(k, 1)))), gel(HM, gtos(k)))); sol = gdiv(sol, content(sol)); if (!gequal0(base)) { H = gmul(H, completebasis(sol, gen_1)); aux = gcopy(gel(H, 1)); gel(H, 1) = gcopy(gel(H, gtos(n))); gel(H, gtos(n)) = gneg(aux); p15 = cgetg(4, t_VEC); gel(p15, 1) = gmul(gmul(gtrans(H), G), H); gel(p15, 2) = gcopy(H); gel(p15, 3) = gcopy(gel(H, 1)); p15 = gerepilecopy(ltop, p15); return p15; } else { p16 = gmul(H, sol); p16 = gerepilecopy(ltop, p16); return p16; } } /* Reduction [k,k-1]. */ Mkk1 = gcopy(gcoeff(M, gtos(k), gtos(gsubgs(k, 1)))); bk1new = gadd(gmul(gsqr(Mkk1), gcoeff(A, gtos(gsubgs(k, 1)), gtos(gsubgs(k, 1)))), gcoeff(A, gtos(k), gtos(k))); if (!gequal0(swap = stoi(gcmp(gabs(bk1new, prec), gmul(c, gabs(gcoeff(A, gtos(gsubgs(k, 1)), gtos(gsubgs(k, 1))), prec))) < 0))) Mkk1new = gdiv(gmul(gneg(Mkk1), gcoeff(A, gtos(gsubgs(k, 1)), gtos(gsubgs(k, 1)))), bk1new); /* Update the matrices after the swap. */ if (!gequal0(swap)) { { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN j = gen_0; for (j = gen_1; gcmp(j, n) <= 0; j = gaddgs(j, 1)) { aux = gcopy(gcoeff(H, gtos(j), gtos(gsubgs(k, 1)))); gcoeff(H, gtos(j), gtos(gsubgs(k, 1))) = gcopy(gcoeff(H, gtos(j), gtos(k))); gcoeff(H, gtos(j), gtos(k)) = gneg(aux); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 3, &j, &aux, &H); } } p17 = gsubgs(k, 2); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN j = gen_0; for (j = gen_1; gcmp(j, p17) <= 0; j = gaddgs(j, 1)) { aux = gcopy(gcoeff(M, gtos(gsubgs(k, 1)), gtos(j))); gcoeff(M, gtos(gsubgs(k, 1)), gtos(j)) = gcopy(gcoeff(M, gtos(k), gtos(j))); gcoeff(M, gtos(k), gtos(j)) = gneg(aux); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 3, &j, &aux, &M); } } p18 = gaddgs(k, 1); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN j = gen_0; for (j = p18; gcmp(j, n) <= 0; j = gaddgs(j, 1)) { aux = gcopy(gcoeff(M, gtos(j), gtos(k))); gcoeff(M, gtos(j), gtos(k)) = gadd(gneg(gcoeff(M, gtos(j), gtos(gsubgs(k, 1)))), gmul(Mkk1, aux)); gcoeff(M, gtos(j), gtos(gsubgs(k, 1))) = gadd(aux, gmul(Mkk1new, gcoeff(M, gtos(j), gtos(k)))); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 3, &j, &aux, &M); } } { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN j = gen_0; for (j = gen_1; gcmp(j, n) <= 0; j = gaddgs(j, 1)) { if (!gequal(j, k) && !gequal(j, gsubgs(k, 1))) { aux = gcopy(gcoeff(A, gtos(gsubgs(k, 1)), gtos(j))); gcoeff(A, gtos(gsubgs(k, 1)), gtos(j)) = gcopy(gcoeff(A, gtos(k), gtos(j))); gcoeff(A, gtos(k), gtos(j)) = gneg(aux); aux = gcopy(gcoeff(A, gtos(j), gtos(gsubgs(k, 1)))); gcoeff(A, gtos(j), gtos(gsubgs(k, 1))) = gadd(gmul(Mkk1, aux), gcoeff(A, gtos(j), gtos(k))); gcoeff(A, gtos(j), gtos(k)) = gsub(gneg(aux), gmul(Mkk1new, gcoeff(A, gtos(j), gtos(gsubgs(k, 1))))); } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 3, &j, &aux, &A); } } aux1 = gcopy(gcoeff(A, gtos(gsubgs(k, 1)), gtos(gsubgs(k, 1)))); aux2 = gcopy(gcoeff(A, gtos(k), gtos(gsubgs(k, 1)))); gcoeff(A, gtos(k), gtos(gsubgs(k, 1))) = gsub(gneg(gcoeff(A, gtos(gsubgs(k, 1)), gtos(k))), gmul(Mkk1, aux1)); gcoeff(A, gtos(gsubgs(k, 1)), gtos(gsubgs(k, 1))) = gadd(gcoeff(A, gtos(k), gtos(k)), gmul(Mkk1, aux2)); gcoeff(A, gtos(k), gtos(k)) = gsub(aux1, gmul(Mkk1new, gcoeff(A, gtos(k), gtos(gsubgs(k, 1))))); gcoeff(A, gtos(gsubgs(k, 1)), gtos(k)) = gsub(gneg(aux2), gmul(Mkk1new, gcoeff(A, gtos(gsubgs(k, 1)), gtos(gsubgs(k, 1))))); gcoeff(M, gtos(k), gtos(gsubgs(k, 1))) = gcopy(Mkk1new); if (gcmpgs(DEBUGLEVEL_qfsolve, 4) >= 0) { newG = gmul(gmul(gtrans(H), G), H); { long i; p19 = cgetg(gtos(n)+1, t_VEC); for (i = 1; gcmpsg(i, n) <= 0; ++i) gel(p19, i) = det(extract0(newG, subis(shifti(gen_1, i), 1), subis(shifti(gen_1, i), 1))); } pari_printf("%Ps\n", p19); } if (!gequalgs(k, 2)) k = gsubgs(k, 1); } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 22, &swap, &q, &p13, &M, &A, &H, &di, &HM, &aux1, &aux2, &sol, &aux, &p15, &p16, &Mkk1, &bk1new, &Mkk1new, &p17, &p18, &newG, &p19, &k); } } p12 = gsubgs(k, 2); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN l = gen_0; long l23 = -1 > 0; /* bool */ GEN p24 = gen_0; for (l = p12; l23?gcmpgs(l, 1) <= 0:gcmpgs(l, 1) >= 0; l = gaddgs(l, -1)) { /* red(k,l) */ if (!gequal0(q = ground(gcoeff(M, gtos(k), gtos(l))))) { p24 = gsubgs(l, 1); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN i = gen_0; for (i = gen_1; gcmp(i, p24) <= 0; i = gaddgs(i, 1)) { gcoeff(M, gtos(k), gtos(i)) = gsub(gcoeff(M, gtos(k), gtos(i)), gmul(q, gcoeff(M, gtos(l), gtos(i)))); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &i, &M); } } gcoeff(M, gtos(k), gtos(l)) = gsub(gcoeff(M, gtos(k), gtos(l)), q); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN i = gen_0; for (i = gen_1; gcmp(i, n) <= 0; i = gaddgs(i, 1)) { gcoeff(A, gtos(k), gtos(i)) = gsub(gcoeff(A, gtos(k), gtos(i)), gmul(q, gcoeff(A, gtos(l), gtos(i)))); gcoeff(H, gtos(i), gtos(k)) = gsub(gcoeff(H, gtos(i), gtos(k)), gmul(q, gcoeff(H, gtos(i), gtos(l)))); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 3, &i, &A, &H); } } } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 6, &l, &q, &p24, &M, &A, &H); } } k = gaddgs(k, 1); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 17, &swap, &q, &M, &A, &H, &di, &HM, &aux1, &aux2, &sol, &aux, &Mkk1, &bk1new, &Mkk1new, &newG, &k, &p12); } } p3 = cgetg(3, t_VEC); gel(p3, 1) = gmul(gmul(gtrans(H), G), H); gel(p3, 2) = gcopy(H); p3 = gerepilecopy(ltop, p3); return p3; } GEN kermodp(GEN M, GEN p) /* vec */ { pari_sp ltop = avma; GEN n = gen_0, U = gen_0, d = gen_0; GEN p1 = gen_0, p2 = gen_0; /* vec */ n = stoi(glength(M)); U = centerlift(matker0(gmul(M, gmodulsg(1, p)), 0)); d = stoi(glength(U)); U = completebasis(U, NULL); { long i, j; p1 = cgetg(gtos(n)+1, t_MAT); for (j = 1; gcmpsg(j, n) <= 0; ++j) { gel(p1, j) = cgetg(gtos(n)+1, t_COL); for (i = 1; gcmpsg(i, n) <= 0; ++i) gcoeff(p1, i, j) = gcopy(gcoeff(U, i, gtos(gsubgs(gaddgs(n, 1), j)))); } } U = p1; p2 = cgetg(3, t_VEC); gel(p2, 1) = gcopy(d); gel(p2, 2) = gcopy(U); p2 = gerepilecopy(ltop, p2); return p2; } GEN Qfparam(GEN G, GEN sol, GEN fl, long prec) { pari_sp ltop = avma; GEN U = gen_0, G1 = gen_0, G2 = gen_0; GEN p1 = gen_0, p2 = gen_0, p3 = gen_0; /* vec */ if (!fl) fl = stoi(3); if (gcmpgs(DEBUGLEVEL_qfsolve, 5) >= 0) pari_printf(" starting Qfparam\n"); sol = gdiv(sol, content(sol)); /* build U such that U[,3] = sol, and det(U) = +-1 */ U = completebasis(sol, gen_1); G1 = gmul(gmul(gtrans(U), G), U); p1 = cgetg(4, t_MAT); gel(p1, 1) = cgetg(4, t_COL); gel(p1, 2) = cgetg(4, t_COL); gel(p1, 3) = cgetg(4, t_COL); gcoeff(p1, 1, 1) = gmulsg(-2, gcoeff(G1, 1, 3)); gcoeff(p1, 1, 2) = gmulsg(-2, gcoeff(G1, 2, 3)); gcoeff(p1, 1, 3) = gen_0; gcoeff(p1, 2, 1) = gen_0; gcoeff(p1, 2, 2) = gmulsg(-2, gcoeff(G1, 1, 3)); gcoeff(p1, 2, 3) = gmulsg(-2, gcoeff(G1, 2, 3)); gcoeff(p1, 3, 1) = gcopy(gcoeff(G1, 1, 1)); gcoeff(p1, 3, 2) = gmulsg(2, gcoeff(G1, 1, 2)); gcoeff(p1, 3, 3) = gcopy(gcoeff(G1, 2, 2)); /* G1 has a 0 at the bottom right corner */ G2 = p1; sol = gmul(U, G2); p2 = cgetg(3, t_MAT); gel(p2, 1) = cgetg(3, t_COL); gel(p2, 2) = cgetg(3, t_COL); gcoeff(p2, 1, 1) = gcopy(gcoeff(sol, gtos(fl), 1)); gcoeff(p2, 1, 2) = gdivgs(gcoeff(sol, gtos(fl), 2), 2); gcoeff(p2, 2, 1) = gdivgs(gcoeff(sol, gtos(fl), 2), 2); gcoeff(p2, 2, 2) = gcopy(gcoeff(sol, gtos(fl), 3)); if (!gequal0(fl) && (gequal0(gissquare(gneg(det(U = p2)))))) { U = QfbReduce(U, prec); p3 = cgetg(4, t_MAT); gel(p3, 1) = cgetg(4, t_COL); gel(p3, 2) = cgetg(4, t_COL); gel(p3, 3) = cgetg(4, t_COL); gcoeff(p3, 1, 1) = gsqr(gcoeff(U, 1, 1)); gcoeff(p3, 1, 2) = gmul(gmulsg(2, gcoeff(U, 1, 2)), gcoeff(U, 1, 1)); gcoeff(p3, 1, 3) = gsqr(gcoeff(U, 1, 2)); gcoeff(p3, 2, 1) = gmul(gcoeff(U, 2, 1), gcoeff(U, 1, 1)); gcoeff(p3, 2, 2) = gadd(gmul(gcoeff(U, 2, 2), gcoeff(U, 1, 1)), gmul(gcoeff(U, 2, 1), gcoeff(U, 1, 2))); gcoeff(p3, 2, 3) = gmul(gcoeff(U, 1, 2), gcoeff(U, 2, 2)); gcoeff(p3, 3, 1) = gsqr(gcoeff(U, 2, 1)); gcoeff(p3, 3, 2) = gmul(gmulsg(2, gcoeff(U, 2, 1)), gcoeff(U, 2, 2)); gcoeff(p3, 3, 3) = gsqr(gcoeff(U, 2, 2)); U = p3; sol = gmul(sol, U); } if (gcmpgs(DEBUGLEVEL_qfsolve, 5) >= 0) pari_printf(" end of Qfparam\n"); sol = gerepilecopy(ltop, sol); return sol; } GEN LLLgoon3(GEN G, GEN c, long prec) /* vec */ { pari_sp ltop = avma; GEN red = gen_0, U1 = gen_0, G2 = gen_0, bez = gen_0, U2 = gen_0, G3 = gen_0, cc = gen_0, U3 = gen_0; GEN p1 = gen_0, p2 = gen_0, p3 = gen_0, p4 = gen_0; /* vec */ if (!c) c = gen_1; red = IndefiniteLLL(G, c, gen_1, prec); p1 = cgetg(4, t_MAT); gel(p1, 1) = cgetg(4, t_COL); gel(p1, 2) = cgetg(4, t_COL); gel(p1, 3) = cgetg(4, t_COL); gcoeff(p1, 1, 1) = gen_0; gcoeff(p1, 1, 2) = gen_0; gcoeff(p1, 1, 3) = gen_1; gcoeff(p1, 2, 1) = gen_0; gcoeff(p1, 2, 2) = gen_1; gcoeff(p1, 2, 3) = gen_0; gcoeff(p1, 3, 1) = gen_1; gcoeff(p1, 3, 2) = gen_0; gcoeff(p1, 3, 3) = gen_0; /* We always find an isotropic vector. */ U1 = p1; G2 = gmul(gmul(gtrans(U1), gel(red, 1)), U1); /* G2 has a 0 at the bottom right corner. */ bez = vecbezout(gcoeff(G2, 3, 1), gcoeff(G2, 3, 2)); p2 = cgetg(4, t_MAT); gel(p2, 1) = cgetg(4, t_COL); gel(p2, 2) = cgetg(4, t_COL); gel(p2, 3) = cgetg(4, t_COL); gcoeff(p2, 1, 1) = gcopy(gel(bez, 1)); gcoeff(p2, 1, 2) = gdiv(gcoeff(G2, 3, 2), gel(bez, 3)); gcoeff(p2, 1, 3) = gen_0; gcoeff(p2, 2, 1) = gcopy(gel(bez, 2)); gcoeff(p2, 2, 2) = gdiv(gneg(gcoeff(G2, 3, 1)), gel(bez, 3)); gcoeff(p2, 2, 3) = gen_0; gcoeff(p2, 3, 1) = gen_0; gcoeff(p2, 3, 2) = gen_0; gcoeff(p2, 3, 3) = gen_m1; U2 = p2; G3 = gmul(gmul(gtrans(U2), G2), U2); /* G3 has 0 under the co-diagonal. */ cc = gmodgs(gcoeff(G3, 1, 1), 2); p3 = cgetg(4, t_MAT); gel(p3, 1) = cgetg(4, t_COL); gel(p3, 2) = cgetg(4, t_COL); gel(p3, 3) = cgetg(4, t_COL); gcoeff(p3, 1, 1) = gen_1; gcoeff(p3, 1, 2) = gen_0; gcoeff(p3, 1, 3) = gen_0; gcoeff(p3, 2, 1) = gcopy(cc); gcoeff(p3, 2, 2) = gen_1; gcoeff(p3, 2, 3) = gen_0; gcoeff(p3, 3, 1) = ground(gdiv(gdivgs(gneg(gadd(gcoeff(G3, 1, 1), gmul(cc, gadd(gmulsg(2, gcoeff(G3, 1, 2)), gmul(gcoeff(G3, 2, 2), cc))))), 2), gcoeff(G3, 1, 3))); gcoeff(p3, 3, 2) = ground(gdiv(gneg(gadd(gcoeff(G3, 1, 2), gmul(cc, gcoeff(G3, 2, 2)))), gcoeff(G3, 1, 3))); gcoeff(p3, 3, 3) = gen_1; U3 = p3; p4 = cgetg(3, t_VEC); gel(p4, 1) = gmul(gmul(gtrans(U3), G3), U3); gel(p4, 2) = gmul(gmul(gmul(gel(red, 2), U1), U2), U3); p4 = gerepilecopy(ltop, p4); return p4; } GEN completebasis(GEN v, GEN redflag) { pari_sp ltop = avma; GEN U = gen_0, n = gen_0, re = gen_0; GEN p1 = gen_0; /* vec */ GEN p2 = gen_0; if (!redflag) redflag = gen_0; v = gtomat(v); n = stoi(glength(gtrans(v))); if (gequalgs(n, glength(v))) { v = gerepilecopy(ltop, v); return v; } U = ginv(gtrans(gel(mathnf0(gtrans(v), 1), 2))); if (gequal1(n) || (gequal0(redflag))) { U = gerepilecopy(ltop, U); return U; } re = lll(extract0(U, subis(shifti(gen_1, gtos(n)), 1), subis(shifti(gen_1, gtos(gsubgs(n, glength(v)))), 1))); p1 = cgetg(3, t_VEC); gel(p1, 1) = gcopy(re); gel(p1, 2) = matid(glength(v)); p2 = gmul(U, matdiagonalblock(p1)); p2 = gerepilecopy(ltop, p2); return p2; } GEN LLLgoon(GEN G, GEN c, long prec) { pari_sp ltop = avma; GEN red = gen_0, U1 = gen_0, G2 = gen_0, U2 = gen_0, G3 = gen_0, n = gen_0, U3 = gen_0, G4 = gen_0, U = gen_0, V = gen_0, B = gen_0, U4 = gen_0, G5 = gen_0, U5 = gen_0, G6 = gen_0; GEN p1 = gen_0, p2 = gen_0, p3 = gen_0, p4 = gen_0; /* vec */ GEN p5 = gen_0; GEN p6 = gen_0; /* vec */ GEN p7 = gen_0, p8 = gen_0; GEN p9 = gen_0, p10 = gen_0, p11 = gen_0; /* vec */ if (!c) c = gen_1; red = IndefiniteLLL(G, c, gen_1, prec); /* If no isotropic vector is found, nothing to do. */ if (glength(red) == 2) { red = gerepilecopy(ltop, red); return red; } /* otherwise: */ U1 = gcopy(gel(red, 2)); G2 = gcopy(gel(red, 1)); /* On a G2[1,1] = 0 */ U2 = gcopy(gel(mathnf0(gtomat(rowcopy(G2, 1)), 4), 2)); G3 = gmul(gmul(gtrans(U2), G2), U2); /* The first line of the matrix G3 only contains 0, */ /* except some 'g' on the right, where g^2| det G. */ n = stoi(glength(G)); U3 = matid(gtos(n)); gcoeff(U3, 1, gtos(n)) = ground(gdivgs(gdiv(gneg(gcoeff(G3, gtos(n), gtos(n))), gcoeff(G3, 1, gtos(n))), 2)); G4 = gmul(gmul(gtrans(U3), G3), U3); p1 = cgetg(3, t_VEC); gel(p1, 1) = gen_1; gel(p1, 2) = gcopy(n); p2 = cgetg(3, t_VEC); gel(p2, 1) = gen_1; gel(p2, 2) = gcopy(n); /* The coeff G4[n,n] is reduced modulo 2g */ U = extract0(G4, p1, p2); if (gequalgs(n, 2)) { { long l12, l13; p3 = cgetg(1, t_MAT); for (l13 = 1; l13 <= 0; ++l13) { gel(p3, l13) = cgetg(3, t_COL); for (l12 = 1; l12 <= 2; ++l12) gcoeff(p3, l12, l13) = gen_0; } } V = p3; } else { p4 = cgetg(3, t_VEC); gel(p4, 1) = gen_1; gel(p4, 2) = gcopy(n); V = extract0(G4, p4, subis(shifti(gen_1, gtos(gsubgs(n, 1))), 2)); } B = ground(gmul(gneg(ginv(U)), V)); U4 = matid(gtos(n)); p5 = gsubgs(n, 1); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN j = gen_0; for (j = gen_2; gcmp(j, p5) <= 0; j = gaddgs(j, 1)) { gcoeff(U4, 1, gtos(j)) = gcopy(gcoeff(B, 1, gtos(gsubgs(j, 1)))); gcoeff(U4, gtos(n), gtos(j)) = gcopy(gcoeff(B, 2, gtos(gsubgs(j, 1)))); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &j, &U4); } } G5 = gmul(gmul(gtrans(U4), G4), U4); /* The last column of G5 is reduced */ if (gcmpgs(n, 4) < 0) { p6 = cgetg(3, t_VEC); gel(p6, 1) = gcopy(G5); gel(p6, 2) = gmul(gmul(gmul(U1, U2), U3), U4); p6 = gerepilecopy(ltop, p6); return p6; } p7 = gsubgs(n, 2); p8 = gsubgs(n, 2); { long i, j; p9 = cgetg(gtos(p7)+1, t_MAT); for (j = 1; gcmpsg(j, p7) <= 0; ++j) { gel(p9, j) = cgetg(gtos(p8)+1, t_COL); for (i = 1; gcmpsg(i, p8) <= 0; ++i) gcoeff(p9, i, j) = gcopy(gcoeff(G5, i + 1, j + 1)); } } red = LLLgoon(p9, c, prec); p10 = cgetg(4, t_VEC); gel(p10, 1) = gtomat(gen_1); gel(p10, 2) = gcopy(gel(red, 2)); gel(p10, 3) = gtomat(gen_1); U5 = matdiagonalblock(p10); G6 = gmul(gmul(gtrans(U5), G5), U5); p11 = cgetg(3, t_VEC); gel(p11, 1) = gcopy(G6); gel(p11, 2) = gmul(gmul(gmul(gmul(U1, U2), U3), U4), U5); p11 = gerepilecopy(ltop, p11); return p11; } GEN QfWittinvariant(GEN G, GEN p) { pari_sp ltop = avma; GEN n = gen_0, det_s = gen_0, diag = gen_0, c = gen_0, p1 = gen_0; GEN p2 = gen_0, p3 = gen_0; /* vec */ GEN p4 = gen_0; n = stoi(glength(G)); p1 = gaddgs(n, 1); { long i, l5, l6; GEN p7 = gen_0; /* vec */ p2 = cgetg(gtos(p1)+1, t_VEC); for (i = 1; gcmpsg(i, p1) <= 0; ++i) { l5 = i - 1; l6 = i - 1; { long j, k; p7 = cgetg(l5+1, t_MAT); for (k = 1; k <= l5; ++k) { gel(p7, k) = cgetg(l6+1, t_COL); for (j = 1; j <= l6; ++j) gcoeff(p7, j, k) = gcopy(gcoeff(G, j, k)); } } gel(p2, i) = det(p7); } } /* Diagonalize G first. */ det_s = p2; { long i; p3 = cgetg(gtos(n)+1, t_VEC); for (i = 1; gcmpsg(i, n) <= 0; ++i) gel(p3, i) = gdiv(gel(det_s, i + 1), gel(det_s, i)); } diag = p3; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN i = gen_0, p8 = gen_0, p9 = gen_0; p4 = gen_1; for (i = gen_1; gcmp(i, n) <= 0; i = gaddgs(i, 1)) { p8 = gaddgs(i, 1); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN j = gen_0; p9 = gen_1; for (j = p8; gcmp(j, n) <= 0; j = gaddgs(j, 1)) { p9 = gmulgs(p9, hilbert(gel(diag, gtos(i)), gel(diag, gtos(j)), p)); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &p9, &j); } } p4 = gmul(p4, p9); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 4, &p4, &i, &p8, &p9); } } /* Then compute Hilbert symbols */ c = p4; c = gerepilecopy(ltop, c); return c; } GEN Qflisteinvariants(GEN G, GEN fa, long prec) /* vec */ { pari_sp ltop = avma; GEN l = gen_0, sol = gen_0, n = gen_0, det_s = gen_0; GEN p1 = gen_0; /* vec */ long l2; GEN p3 = gen_0, p4 = gen_0; /* vec */ long l5; GEN p6 = gen_0, p7 = gen_0; /* vec */ if (!fa) fa = cgetg(1, t_VEC); G = gcopy(G); if (gcmpgs(DEBUGLEVEL_qfsolve, 4) >= 0) pari_printf(" starting Qflisteinvariants %Ps\n", G); if (typ(G) != t_VEC) { p1 = cgetg(2, t_VEC); gel(p1, 1) = gcopy(G); G = p1; } l = stoi(glength(G)); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN j = gen_0; for (j = gen_1; gcmp(j, l) <= 0; j = gaddgs(j, 1)) { if ((typ(gel(G, gtos(j))) == t_QFI) || (typ(gel(G, gtos(j))) == t_QFR)) gel(G, gtos(j)) = mymat(gel(G, gtos(j))); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &j, &G); } } if (!glength(fa)) fa = gcopy(gel(factor(gneg(gabs(gmulsg(2, det(gel(G, 1))), prec))), 1)); if (glength(gel(G, 1)) == 2) { /* In dimension 2, each invariant is a single Hilbert symbol. */ det_s = gneg(det(gel(G, 1))); l2 = glength(fa); { long i, j; p3 = cgetg(gtos(l)+1, t_MAT); for (j = 1; gcmpsg(j, l) <= 0; ++j) { gel(p3, j) = cgetg(l2+1, t_COL); for (i = 1; i <= l2; ++i) gcoeff(p3, i, j) = stoi(hilbert(gcoeff(gel(G, j), 1, 1), det_s, gel(fa, i)) < 0); } } sol = p3; if (gcmpgs(DEBUGLEVEL_qfsolve, 4) >= 0) pari_printf(" end of Qflisteinvariants\n"); p4 = cgetg(3, t_VEC); gel(p4, 1) = gcopy(fa); gel(p4, 2) = gcopy(sol); p4 = gerepilecopy(ltop, p4); return p4; } l5 = glength(fa); { long l8, l9; p6 = cgetg(gtos(l)+1, t_MAT); for (l9 = 1; gcmpsg(l9, l) <= 0; ++l9) { gel(p6, l9) = cgetg(l5+1, t_COL); for (l8 = 1; l8 <= l5; ++l8) gcoeff(p6, l8, l9) = gen_0; } } sol = p6; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN j = gen_0, p10 = gen_0; GEN p11 = gen_0; /* vec */ long l12; for (j = gen_1; gcmp(j, l) <= 0; j = gaddgs(j, 1)) { n = stoi(glength(gel(G, gtos(j)))); p10 = gaddgs(n, 1); { long i, l13, l14; GEN p15 = gen_0; /* vec */ p11 = cgetg(gtos(p10)+1, t_VEC); for (i = 1; gcmpsg(i, p10) <= 0; ++i) { l13 = i - 1; l14 = i - 1; { long k, m; p15 = cgetg(l13+1, t_MAT); for (m = 1; m <= l13; ++m) { gel(p15, m) = cgetg(l14+1, t_COL); for (k = 1; k <= l14; ++k) gcoeff(p15, k, m) = gcopy(gcoeff(gel(G, gtos(j)), k, m)); } } gel(p11, i) = det(p15); } } /* In dimension n, we need to compute a product of n Hilbert symbols. */ det_s = p11; l12 = glength(fa); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; GEN p16 = gen_0, p17 = gen_0; for (i = 1; i <= l12; ++i) { p16 = gsubgs(n, 1); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN k = gen_0; long l18; p17 = gen_1; for (k = gen_1; gcmp(k, p16) <= 0; k = gaddgs(k, 1)) { l18 = gtos(gaddgs(k, 1)); p17 = gmulgs(p17, hilbert(gneg(gel(det_s, gtos(k))), gel(det_s, l18), gel(fa, i))); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &p17, &k); } } gcoeff(sol, i, gtos(j)) = stoi(gcmpgs(gmulgs(p17, hilbert(gel(det_s, gtos(n)), gel(det_s, gtos(gaddgs(n, 1))), gel(fa, i))), 0) < 0); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 3, &p16, &p17, &sol); } } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 6, &j, &n, &p10, &p11, &det_s, &sol); } } if (gcmpgs(DEBUGLEVEL_qfsolve, 4) >= 0) pari_printf(" end of Qflisteinvariants\n"); p7 = cgetg(3, t_VEC); gel(p7, 1) = gcopy(fa); gel(p7, 2) = gcopy(sol); p7 = gerepilecopy(ltop, p7); return p7; } GEN Qfsolvemodp(GEN G, GEN p, long prec) { pari_sp ltop = avma; GEN n = gen_0, vdet = gen_0, G2 = gen_0, sol = gen_0, x1 = gen_0, x2 = gen_0, x3 = gen_0, N1 = gen_0, N2 = gen_0, N3 = gen_0, s = gen_0, r = gen_0; GEN p1 = gen_0, p2 = gen_0, p3 = gen_0, p4 = gen_0, p5 = gen_0; /* vec */ n = stoi(glength(G)); p1 = cgetg(4, t_VEC); gel(p1, 1) = gen_0; gel(p1, 2) = gen_0; gel(p1, 3) = gen_0; vdet = p1; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; GEN p6 = gen_0; /* vec */ for (i = 1; i <= 3; ++i) { G2 = gmul(extract0(G, subis(shifti(gen_1, i), 1), subis(shifti(gen_1, i), 1)), gmodulsg(1, p)); if (gequal0(gel(vdet, i) = det(G2))) { sol = gcopy(gel(gel(kermodp(lift(G2), p), 2), 1)); { long j; GEN p7 = gen_0; p6 = cgetg(gtos(n)+1, t_COL); for (j = 1; gcmpsg(j, n) <= 0; ++j) { if (j <= i) p7 = gcopy(gel(sol, j)); gel(p6, j) = p7; } } sol = p6; sol = gerepilecopy(ltop, sol); return sol; } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 4, &G2, &vdet, &sol, &p6); } } p2 = cgetg(4, t_COL); gel(p2, 1) = gen_1; gel(p2, 2) = gen_0; gel(p2, 3) = gen_0; /* now, solve in dimension 3... */ /* reduction to the diagonal case: */ x1 = p2; p3 = cgetg(4, t_COL); gel(p3, 1) = gneg(gcoeff(G2, 1, 2)); gel(p3, 2) = gcopy(gcoeff(G2, 1, 1)); gel(p3, 3) = gen_0; x2 = p3; p4 = cgetg(4, t_COL); gel(p4, 1) = gsub(gmul(gcoeff(G2, 2, 2), gcoeff(G2, 1, 3)), gmul(gcoeff(G2, 2, 3), gcoeff(G2, 1, 2))); gel(p4, 2) = gsub(gmul(gcoeff(G2, 1, 1), gcoeff(G2, 2, 3)), gmul(gcoeff(G2, 1, 3), gcoeff(G2, 1, 2))); gel(p4, 3) = gsub(gsqr(gcoeff(G2, 1, 2)), gmul(gcoeff(G2, 1, 1), gcoeff(G2, 2, 2))); x3 = p4; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); while (1) { if (!gequal0(gissquare(N1 = gneg(gel(vdet, 2))))) { s = gsqrt(N1, prec); sol = gadd(gmul(s, x1), x2); break; } if (!gequal0(gissquare(N2 = gdiv(gneg(gel(vdet, 3)), gel(vdet, 1))))) { s = gsqrt(N2, prec); sol = gadd(gmul(s, x2), x3); break; } if (!gequal0(gissquare(N3 = gdiv(gmul(gneg(gel(vdet, 2)), gel(vdet, 3)), gel(vdet, 1))))) { s = gsqrt(N3, prec); sol = gadd(gmul(s, x1), x3); break; } r = gen_1; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); while (gequal0(gissquare(s = gdiv(gsubsg(1, gmul(N1, gsqr(r))), N3)))) { r = genrand(p); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &s, &r); } } s = gsqrt(s, prec); sol = gadd(gadd(x1, gmul(r, x2)), gmul(s, x3)); break; if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 6, &N1, &s, &sol, &N2, &N3, &r); } } { long j; GEN p8 = gen_0; p5 = cgetg(gtos(n)+1, t_COL); for (j = 1; gcmpsg(j, n) <= 0; ++j) { if (j <= 3) p8 = gcopy(gel(sol, j)); gel(p5, j) = p8; } } sol = p5; sol = gerepilecopy(ltop, sol); return sol; } GEN Qfminim(GEN G, GEN factdetG, long prec) { pari_sp ltop = avma; GEN n = gen_0, factd = gen_0, detG = gen_0, i = gen_0, U = gen_0, vp = gen_0, Ker = gen_0, dimKer = gen_0, Ker2 = gen_0, dimKer2 = gen_0, sol = gen_0, aux = gen_0, p = gen_0, di = gen_0, m = gen_0; GEN p1 = gen_0, p2 = gen_0; /* vec */ if (!factdetG) factdetG = gen_0; factdetG = gcopy(factdetG); n = stoi(glength(G)); { long l3, l4; p1 = cgetg(3, t_MAT); for (l4 = 1; l4 <= 2; ++l4) { gel(p1, l4) = cgetg(1, t_COL); for (l3 = 1; l3 <= 0; ++l3) gcoeff(p1, l3, l4) = gen_0; } } factd = p1; if (gequal0(factdetG)) { detG = det(G); factdetG = factor(detG); } i = gen_1; U = matid(gtos(n)); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN p5 = gen_0, p6 = gen_0, p7 = gen_0, p8 = gen_0, p9 = gen_0, p10 = gen_0, p11 = gen_0; /* vec */ GEN p12 = gen_0; GEN p13 = gen_0; /* vec */ while (gcmpgs(i, glength(gel(factdetG, 1))) <= 0) { p = gcopy(gcoeff(factdetG, gtos(i), 1)); if (gequalm1(p)) { i = gaddgs(i, 1); continue; } vp = gcopy(gcoeff(factdetG, gtos(i), 2)); if (gcmpgs(DEBUGLEVEL_qfsolve, 4) >= 0) pari_printf(" p = %Ps^%Ps\n", p, vp); if (gequal0(vp)) { i = gaddgs(i, 1); continue; } /* The case vp = 1 can be minimized only if n is odd. */ if (gequal1(vp) && gequal0(gmodgs(n, 2))) { p5 = cgetg(3, t_VEC); gel(p5, 1) = gcopy(p); gel(p5, 2) = gen_1; factd = gtrans(concat(gtrans(factd), gtrans(gtomat(p5)))); i = gaddgs(i, 1); continue; } Ker = kermodp(G, p); dimKer = gcopy(gel(Ker, 1)); Ker = gcopy(gel(Ker, 2)); /* Rem: we must have dimKer <= vp */ if (gcmpgs(DEBUGLEVEL_qfsolve, 4) >= 0) pari_printf(" dimKer = %Ps\n", dimKer); /* trivial case: dimKer = n */ if (gequal(dimKer, n)) { if (gcmpgs(DEBUGLEVEL_qfsolve, 4) >= 0) pari_printf(" case 0: dimKer = n\n"); G = gdiv(G, p); gcoeff(factdetG, gtos(i), 2) = gsub(gcoeff(factdetG, gtos(i), 2), n); continue; } G = gmul(gmul(gtrans(Ker), G), Ker); U = gmul(U, Ker); /* 1st case: dimKer < vp */ /* then the kernel mod p contains a kernel mod p^2 */ if (gcmp(dimKer, vp) < 0) { if (gcmpgs(DEBUGLEVEL_qfsolve, 4) >= 0) pari_printf(" case 1: dimker < vp\n"); { long j, k; p6 = cgetg(gtos(dimKer)+1, t_MAT); for (k = 1; gcmpsg(k, dimKer) <= 0; ++k) { gel(p6, k) = cgetg(gtos(dimKer)+1, t_COL); for (j = 1; gcmpsg(j, dimKer) <= 0; ++j) gcoeff(p6, j, k) = gdiv(gcoeff(G, j, k), p); } } Ker2 = kermodp(p6, p); dimKer2 = gcopy(gel(Ker2, 1)); Ker2 = gcopy(gel(Ker2, 2)); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN j = gen_0; for (j = gen_1; gcmp(j, dimKer2) <= 0; j = gaddgs(j, 1)) { gel(Ker2, gtos(j)) = gdiv(gel(Ker2, gtos(j)), p); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &j, &Ker2); } } p7 = cgetg(3, t_VEC); gel(p7, 1) = gcopy(Ker2); gel(p7, 2) = matid(gtos(gsub(n, dimKer))); Ker2 = matdiagonalblock(p7); G = gmul(gmul(gtrans(Ker2), G), Ker2); U = gmul(U, Ker2); gcoeff(factdetG, gtos(i), 2) = gsub(gcoeff(factdetG, gtos(i), 2), gmulsg(2, dimKer2)); if (gcmpgs(DEBUGLEVEL_qfsolve, 4) >= 0) pari_printf(" end of case 1\n"); continue; } /* Now, we have vp = dimKer */ /* 2nd case: the dimension of the kernel is >=2 */ /* and contains an element of norm 0 mod p^2 */ if ((gcmpgs(dimKer, 2) > 0) || (gequalgs(dimKer, 2) && !gequal0(gissquare(di = gmodulo(gdiv(gsub(gsqr(gcoeff(G, 1, 2)), gmul(gcoeff(G, 1, 1), gcoeff(G, 2, 2))), gsqr(p)), p))))) { /* search for an element of norm p^2... in the kernel */ if (gcmpgs(dimKer, 2) > 0) { if (gcmpgs(DEBUGLEVEL_qfsolve, 4) >= 0) pari_printf(" case 2.1\n"); dimKer = stoi(3); { long j, k; p8 = cgetg(4, t_MAT); for (k = 1; k <= 3; ++k) { gel(p8, k) = cgetg(4, t_COL); for (j = 1; j <= 3; ++j) gcoeff(p8, j, k) = gdiv(gcoeff(G, j, k), p); } } sol = Qfsolvemodp(p8, p, prec); } else { if (gcmpgs(DEBUGLEVEL_qfsolve, 4) >= 0) pari_printf(" case 2.2\n"); if (gequal0(gmod(gcoeff(G, 1, 1), gsqr(p)))) { p9 = cgetg(3, t_COL); gel(p9, 1) = gen_1; gel(p9, 2) = gen_0; sol = p9; } else { p10 = cgetg(3, t_COL); gel(p10, 1) = gadd(gdiv(gneg(gcoeff(G, 1, 2)), p), gsqrt(di, prec)); gel(p10, 2) = gmodulo(gdiv(gcoeff(G, 1, 1), p), p); sol = p10; } } sol = centerlift(sol); sol = gdiv(sol, content(sol)); if (gcmpgs(DEBUGLEVEL_qfsolve, 4) >= 0) pari_printf(" sol = %Ps\n", sol); { long j; GEN p14 = gen_0; p11 = cgetg(gtos(n)+1, t_COL); for (j = 1; gcmpsg(j, n) <= 0; ++j) { if (gcmpsg(j, dimKer) <= 0) p14 = gcopy(gel(sol, j)); else p14 = gen_0; gel(p11, j) = p14; } } Ker = p11; /* fill with 0's */ Ker = completebasis(Ker, gen_1); gel(Ker, gtos(n)) = gdiv(gel(Ker, gtos(n)), p); G = gmul(gmul(gtrans(Ker), G), Ker); U = gmul(U, Ker); gcoeff(factdetG, gtos(i), 2) = gsubgs(gcoeff(factdetG, gtos(i), 2), 2); if (gcmpgs(DEBUGLEVEL_qfsolve, 4) >= 0) pari_printf(" end of case 2\n"); continue; } /* Now, we have vp = dimKer <= 2 */ /* and the kernel contains no vector with norm p^2... */ /* In some cases, exchanging the kernel and the image */ /* makes the minimization easy. */ m = gsubgs(gdiventgs(gsubgs(n, 1), 2), 1); if (((gequal1(vp) && !gequal0(gissquare(gmodulo(gdiv(gmul(gneg(gpow(gen_m1, m, prec)), det(G)), gcoeff(G, 1, 1)), p)))) || ((gequalgs(vp, 2) && gequal1(gmodgs(n, 2))) && (gcmpgs(n, 5) >= 0))) || ((gequalgs(vp, 2) && gequal0(gmodgs(n, 2))) && (gequal0(gissquare(gmodulo(gdiv(gmul(gpow(gen_m1, m, prec), det(G)), gsqr(p)), p)))))) { if (gcmpgs(DEBUGLEVEL_qfsolve, 4) >= 0) pari_printf(" case 3\n"); Ker = matid(gtos(n)); p12 = gaddgs(dimKer, 1); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN j = gen_0; for (j = p12; gcmp(j, n) <= 0; j = gaddgs(j, 1)) { gcoeff(Ker, gtos(j), gtos(j)) = gcopy(p); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &j, &Ker); } } G = gdiv(gmul(gmul(gtrans(Ker), G), Ker), p); U = gmul(U, Ker); gcoeff(factdetG, gtos(i), 2) = gsub(gcoeff(factdetG, gtos(i), 2), gsub(gmulsg(2, dimKer), n)); if (gcmpgs(DEBUGLEVEL_qfsolve, 4) >= 0) pari_printf(" end of case 3\n"); continue; } /* Minimization was not possible se far. */ /* If n == 3 or 4, this proves the local non-solubility at p. */ if (gequalgs(n, 3) || gequalgs(n, 4)) { if (gcmpgs(DEBUGLEVEL_qfsolve, 1) >= 0) pari_printf(" no local solution at %Ps\n", p); p = gerepilecopy(ltop, p); return p; } if (gcmpgs(DEBUGLEVEL_qfsolve, 4) >= 0) pari_printf(" prime %Ps finished\n", p); p13 = cgetg(3, t_VEC); gel(p13, 1) = gcopy(p); gel(p13, 2) = gcopy(vp); factd = gtrans(concat(gtrans(factd), gtrans(gtomat(p13)))); i = gaddgs(i, 1); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 23, &p, &i, &vp, &p5, &factd, &Ker, &dimKer, &G, &factdetG, &U, &p6, &Ker2, &dimKer2, &p7, &di, &p8, &sol, &p9, &p10, &p11, &m, &p12, &p13); } } /* apply LLL to avoid coefficients explosion */ aux = lll(U); p2 = cgetg(4, t_VEC); gel(p2, 1) = gmul(gmul(gtrans(aux), G), aux); gel(p2, 2) = gmul(U, aux); gel(p2, 3) = gcopy(factd); p2 = gerepilecopy(ltop, p2); return p2; } GEN mymat(GEN qfb) /* vec */ { pari_sp ltop = avma; GEN p1 = gen_0; /* vec */ qfb = gtovec(qfb); p1 = cgetg(3, t_MAT); gel(p1, 1) = cgetg(3, t_COL); gel(p1, 2) = cgetg(3, t_COL); gcoeff(p1, 1, 1) = gcopy(gel(qfb, 1)); gcoeff(p1, 1, 2) = gdivgs(gel(qfb, 2), 2); gcoeff(p1, 2, 1) = gdivgs(gel(qfb, 2), 2); gcoeff(p1, 2, 2) = gcopy(gel(qfb, 3)); p1 = gerepilecopy(ltop, p1); return p1; } GEN Qfbsqrtgauss(GEN G, GEN factdetG, long prec) { pari_sp ltop = avma; GEN a = gen_0, b = gen_0, c = gen_0, d = gen_0, m = gen_0, n = gen_0, p = gen_0, aux = gen_0, Q1 = gen_0, M = gen_0; long l1; GEN p2 = gen_0; /* vec */ GEN p3 = gen_0, p4 = gen_0; factdetG = gcopy(factdetG); if (gcmpgs(DEBUGLEVEL_qfsolve, 3) >= 0) pari_printf(" starting Qfbsqrtgauss with %Ps%Ps\n", G, factdetG); G = gtovec(G); a = gcopy(gel(G, 1)); b = gdivgs(gel(G, 2), 2); c = gcopy(gel(G, 3)); d = gsub(gmul(a, c), gsqr(b)); /* 1st step: solve m^2 = a (d), m*n = -b (d), n^2 = c (d) */ m = n = gmodulss(1, 1); gcoeff(factdetG, 1, 2) = gsubgs(gcoeff(factdetG, 1, 2), 3); l1 = glength(gel(factdetG, 1)); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; for (i = 1; i <= l1; ++i) { if (gequal0(gcoeff(factdetG, i, 2))) continue; p = gcopy(gcoeff(factdetG, i, 1)); if (gequal1(ggcd(a, p))) { aux = gsqrt(gmodulo(a, p), prec); m = chinese(m, aux); n = chinese(n, gdiv(gneg(b), aux)); } else { aux = gsqrt(gmodulo(c, p), prec); n = chinese(n, aux); m = chinese(m, gdiv(gneg(b), aux)); } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 4, &p, &aux, &m, &n); } } m = centerlift(m); n = centerlift(n); if (gcmpgs(DEBUGLEVEL_qfsolve, 4) >= 0) { pari_printf(" m = %Ps\n", m); pari_printf(" n = %Ps\n", n); } p2 = cgetg(4, t_MAT); gel(p2, 1) = cgetg(4, t_COL); gel(p2, 2) = cgetg(4, t_COL); gel(p2, 3) = cgetg(4, t_COL); gcoeff(p2, 1, 1) = gdiv(gsub(gsqr(n), c), d); gcoeff(p2, 1, 2) = gdiv(gadd(gmul(m, n), b), d); gcoeff(p2, 1, 3) = gcopy(n); gcoeff(p2, 2, 1) = gdiv(gadd(gmul(m, n), b), d); gcoeff(p2, 2, 2) = gdiv(gsub(gsqr(m), a), d); gcoeff(p2, 2, 3) = gcopy(m); gcoeff(p2, 3, 1) = gcopy(n); gcoeff(p2, 3, 2) = gcopy(m); gcoeff(p2, 3, 3) = gcopy(d); /* 2nd step: build Q1, with det=-1 such that Q1(x,y,0) = G(x,y) */ Q1 = p2; Q1 = gneg(matadjoint0(Q1, 0)); /* 3rd step: reduce Q1 to [0,0,-1;0,1,0;-1,0,0] */ M = rowcopy(gel(LLLgoon3(Q1, NULL, prec), 2), 3); if (gcmpgs(gel(M, 1), 0) < 0) M = gneg(M); if (gcmpgs(DEBUGLEVEL_qfsolve, 3) >= 0) pari_printf(" end of Qfbsqrtgauss\n"); if (!gequal0(gmodgs(gel(M, 1), 2))) { p3 = Qfb0(gel(M, 1), gmulsg(2, gel(M, 2)), gmulsg(2, gel(M, 3)), NULL, prec); p3 = gerepilecopy(ltop, p3); return p3; } else { p4 = Qfb0(gel(M, 3), gmulsg(-2, gel(M, 2)), gmulsg(2, gel(M, 1)), NULL, prec); p4 = gerepilecopy(ltop, p4); return p4; } avma = ltop; return gen_0; } GEN class2(GEN D, GEN factdetG, GEN Winvariants, GEN U2, long prec) /* vec */ { pari_sp ltop = avma; GEN factD = gen_0, n = gen_0, rang = gen_0, m = gen_0, listgen = gen_0, vD = gen_0, p = gen_0, vp = gen_0, aux = gen_0, invgen = gen_0, im = gen_0, Ker = gen_0, Kerim = gen_0, listgen2 = gen_0, G2 = gen_0, struct_s = gen_0, E = gen_0, red = gen_0; GEN p1 = gen_0, p2 = gen_0, p3 = gen_0, p4 = gen_0, p5 = gen_0, p6 = gen_0, p7 = gen_0, p8 = gen_0, p9 = gen_0, p10 = gen_0, p11 = gen_0, p12 = gen_0, p13 = gen_0, p14 = gen_0, p15 = gen_0, p16 = gen_0, p17 = gen_0; /* vec */ factdetG = gcopy(factdetG); /* MODI change struct to struct_s */ /* remove compteur */ if (gcmpgs(DEBUGLEVEL_qfsolve, 1) >= 0) pari_printf(" Construction of the 2-class group of discriminant %Ps\n", D); if (gequalgs(gmodgs(D, 4), 2) || gequalgs(gmodgs(D, 4), 3)) pari_err(user, "class2: Discriminant not congruent to 0,1 mod 4"); if (gequalgs(D, -4)) { p1 = cgetg(3, t_VEC); p2 = cgetg(2, t_VEC); gel(p2, 1) = gen_1; gel(p1, 1) = p2; p3 = cgetg(2, t_VEC); gel(p3, 1) = Qfb0(gen_1, gen_0, gen_1, NULL, prec); gel(p1, 2) = p3; p1 = gerepilecopy(ltop, p1); return p1; } if (gequal0(factdetG)) factdetG = factor(gmulsg(2, gabs(D, prec))); p4 = cgetg(2, t_VEC); gel(p4, 1) = gen_m1; factD = concat(p4, gtrans(gel(factdetG, 1))); if (gequal1(gmodgs(D, 4))) { D = gmulgs(D, 4); gcoeff(factdetG, 1, 2) = gaddgs(gcoeff(factdetG, 1, 2), 2); } n = stoi(glength(factD)); rang = gsubgs(n, 3); if (gcmpgs(D, 0) > 0) m = gaddgs(rang, 1); else m = rang; if (gcmpgs(m, 0) < 0) m = gen_0; if (gcmpgs(DEBUGLEVEL_qfsolve, 3) >= 0) pari_printf(" factD = %Ps\n", factD); { long l18; p5 = cgetg(gtos(m)+1, t_VEC); for (l18 = 1; gcmpsg(l18, m) <= 0; ++l18) gel(p5, l18) = gen_0; } listgen = p5; if (!gequal0(vD = stoi(ggval(D, gen_2)))) E = Qfb0(gen_1, gen_0, gdivgs(gneg(D), 4), NULL, prec); else E = Qfb0(gen_1, gen_1, gdivgs(gsubsg(1, D), 4), NULL, prec); if (gcmpgs(DEBUGLEVEL_qfsolve, 3) >= 0) pari_printf(" E = %Ps\n", E); if ((typ(Winvariants) == t_COL) && (gequal0(Winvariants) || (glength(inverseimage(gmul(U2, gmodulss(1, 2)), Winvariants)) > 0))) { p6 = cgetg(3, t_VEC); p7 = cgetg(2, t_VEC); gel(p7, 1) = gen_1; gel(p6, 1) = p7; p8 = cgetg(2, t_VEC); gel(p8, 1) = gcopy(E); gel(p6, 2) = p8; p6 = gerepilecopy(ltop, p6); return p6; } { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN i = gen_0; for (i = gen_1; gcmp(i, m) <= 0; i = gaddgs(i, 1)) { /* no need to look at factD[1]=-1, nor factD[2]=2 */ p = gcopy(gel(factD, gtos(gaddgs(i, 2)))); vp = stoi(ggval(D, p)); aux = gpow(p, vp, prec); if (!gequal0(vD)) gel(listgen, gtos(i)) = Qfb0(aux, gen_0, gdiv(gdivgs(gneg(D), 4), aux), NULL, prec); else gel(listgen, gtos(i)) = Qfb0(aux, aux, gdivgs(gsub(aux, gdiv(D, aux)), 4), NULL, prec); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 5, &i, &p, &vp, &aux, &listgen); } } if (gequalgs(vD, 2) && !gequalgs(gmodgs(D, 16), 4)) { m = gaddgs(m, 1); rang = gaddgs(rang, 1); p9 = cgetg(2, t_VEC); gel(p9, 1) = Qfb0(gen_2, gen_2, gdivgs(gsubsg(4, D), 8), NULL, prec); listgen = concat(listgen, p9); } if (gequalgs(vD, 3)) { m = gaddgs(m, 1); rang = gaddgs(rang, 1); p10 = cgetg(2, t_VEC); gel(p10, 1) = Qfb0(gpow(gen_2, gsubgs(vD, 2), prec), gen_0, gdiv(gneg(D), gpow(gen_2, vD, prec)), NULL, prec); listgen = concat(listgen, p10); } if (gcmpgs(DEBUGLEVEL_qfsolve, 3) >= 0) pari_printf(" listgen = %Ps\n", listgen); if (gcmpgs(DEBUGLEVEL_qfsolve, 2) >= 0) pari_printf(" rank = %Ps\n", rang); if (gequal0(rang)) { p11 = cgetg(3, t_VEC); p12 = cgetg(2, t_VEC); gel(p12, 1) = gen_1; gel(p11, 1) = p12; p13 = cgetg(2, t_VEC); gel(p13, 1) = gcopy(E); gel(p11, 2) = p13; p11 = gerepilecopy(ltop, p11); return p11; } invgen = gmul(gel(Qflisteinvariants(listgen, factD, prec), 2), gmodulss(1, 2)); if (gcmpgs(DEBUGLEVEL_qfsolve, 3) >= 0) pari_printf(" invgen = %Ps\n", lift(invgen)); { long i; p14 = cgetg(gtos(m)+1, t_VEC); for (i = 1; gcmpsg(i, m) <= 0; ++i) gel(p14, i) = gen_2; } /* MODI2.1 */ struct_s = p14; im = lift(inverseimage(invgen, matimage0(invgen, 0))); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN p19 = gen_0; /* vec */ long l20; while ((gcmpsg(glength(im), rang) < 0) || ((typ(Winvariants) == t_COL) && glength(stoi(gequal0(inverseimage(concat(invgen, U2), Winvariants)))))) { Ker = lift(matker0(invgen, 0)); Kerim = concat(Ker, im); { long l21; p19 = cgetg(gtos(m)+1, t_VEC); for (l21 = 1; gcmpsg(l21, m) <= 0; ++l21) gel(p19, l21) = gen_0; } listgen2 = p19; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN i = gen_0; for (i = gen_1; gcmp(i, m) <= 0; i = gaddgs(i, 1)) { gel(listgen2, gtos(i)) = gcopy(E); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN j = gen_0; for (j = gen_1; gcmp(j, m) <= 0; j = gaddgs(j, 1)) { if (!gequal0(gcoeff(Kerim, gtos(j), gtos(i)))) gel(listgen2, gtos(i)) = qfbcompraw(gel(listgen2, gtos(i)), gel(listgen, gtos(j))); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &j, &listgen2); } } if (gcmpgs(gnorml2(gel(Kerim, gtos(i))), 1) > 0) { red = QfbReduce(aux = mymat(gel(listgen2, gtos(i))), prec); aux = gmul(gmul(gtrans(red), aux), red); gel(listgen2, gtos(i)) = Qfb0(gcoeff(aux, 1, 1), gmulsg(2, gcoeff(aux, 1, 2)), gcoeff(aux, 2, 2), NULL, prec); } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 4, &i, &listgen2, &aux, &red); } } listgen = gcopy(listgen2); invgen = gmul(invgen, Kerim); if (gcmpgs(DEBUGLEVEL_qfsolve, 4) >= 0) pari_printf(" listgen = %Ps\n", listgen); if (gcmpgs(DEBUGLEVEL_qfsolve, 4) >= 0) pari_printf(" invgen = %Ps\n", lift(invgen)); /* MODI2.1 */ l20 = glength(Ker); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; for (i = 1; i <= l20; ++i) { G2 = Qfbsqrtgauss(gel(listgen, i), factdetG, prec); gel(struct_s, i) = gshift(gel(struct_s, i), 1); gel(listgen, i) = gcopy(G2); gel(invgen, i) = gmul(gel(gel(Qflisteinvariants(G2, factD, prec), 2), 1), gmodulss(1, 2)); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 4, &G2, &struct_s, &listgen, &invgen); } } if (gcmpgs(DEBUGLEVEL_qfsolve, 3) >= 0) pari_printf(" listgen = %Ps\n", listgen); if (gcmpgs(DEBUGLEVEL_qfsolve, 3) >= 0) pari_printf(" invgen = %Ps\n", lift(invgen)); if (gcmpgs(DEBUGLEVEL_qfsolve, 3) >= 0) pari_printf(" struct = %Ps\n", struct_s); /* MODI2.1 */ im = lift(inverseimage(invgen, matimage0(invgen, 0))); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 11, &Ker, &Kerim, &p19, &listgen2, &aux, &red, &listgen, &invgen, &G2, &struct_s, &im); } } { long l22; p15 = cgetg(gtos(rang)+1, t_VEC); for (l22 = 1; gcmpsg(l22, rang) <= 0; ++l22) gel(p15, l22) = gen_0; } listgen2 = p15; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN i = gen_0; for (i = gen_1; gcmp(i, rang) <= 0; i = gaddgs(i, 1)) { gel(listgen2, gtos(i)) = gcopy(E); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN j = gen_0; for (j = gen_1; gcmp(j, m) <= 0; j = gaddgs(j, 1)) { if (!gequal0(gcoeff(im, gtos(j), gtos(i)))) gel(listgen2, gtos(i)) = qfbcompraw(gel(listgen2, gtos(i)), gel(listgen, gtos(j))); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &j, &listgen2); } } if (gcmpgs(gnorml2(gel(im, gtos(i))), 1) > 0) { red = QfbReduce(aux = mymat(gel(listgen2, gtos(i))), prec); aux = gmul(gmul(gtrans(red), aux), red); gel(listgen2, gtos(i)) = Qfb0(gcoeff(aux, 1, 1), gmulsg(2, gcoeff(aux, 1, 2)), gcoeff(aux, 2, 2), NULL, prec); } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 4, &i, &listgen2, &aux, &red); } } listgen = gcopy(listgen2); { long i; p16 = cgetg(gtos(rang)+1, t_VEC); for (i = 1; gcmpsg(i, rang) <= 0; ++i) gel(p16, i) = gcopy(gel(struct_s, gtos(gaddgs(gsub(m, rang), i)))); } /* listgen = vector(rang,i,listgen[m-rang+i]); */ struct_s = p16; if (gcmpgs(DEBUGLEVEL_qfsolve, 2) >= 0) pari_printf(" listgen = %Ps\n", listgen); if (gcmpgs(DEBUGLEVEL_qfsolve, 2) >= 0) pari_printf(" struct = %Ps\n", struct_s); p17 = cgetg(3, t_VEC); gel(p17, 1) = gcopy(struct_s); gel(p17, 2) = gcopy(listgen); p17 = gerepilecopy(ltop, p17); return p17; } GEN Qfsolve(GEN G, GEN factD, long prec) { pari_sp ltop = avma; GEN n = gen_0, M = gen_0, signG = gen_0, d = gen_0, Min = gen_0, U = gen_0, codim = gen_0, aux = gen_0, G1 = gen_0, detG1 = gen_0, M1 = gen_0, subspace1 = gen_0, G2 = gen_0, subspace2 = gen_0, M2 = gen_0, solG2 = gen_0, Winvariants = gen_0, dQ = gen_0, factd = gen_0, U2 = gen_0, clgp2 = gen_0, V = gen_0, detG2 = gen_0, dimseti = gen_0, solG1 = gen_0, sol = gen_0, Q = gen_0, p1 = gen_0; GEN p2 = gen_0; /* vec */ GEN p3 = gen_0, p4 = gen_0, p5 = gen_0; GEN p6 = gen_0; /* vec */ long l7; GEN p8 = gen_0; GEN p9 = gen_0; /* vec */ GEN p10 = gen_0; GEN p11 = gen_0; /* vec */ GEN p12 = gen_0, p13 = gen_0; GEN p14 = gen_0, p15 = gen_0; /* vec */ long l16; GEN p17 = gen_0; /* vec */ long l18, l19, l20; GEN p21 = gen_0; long l22; GEN p23 = gen_0; /* vec */ long l24; long l25; /* bool */ long l26; GEN p27 = gen_0; /* vec */ long l28; GEN p29 = gen_0; /* vec */ long l30; GEN p31 = gen_0; /* vec */ long l32; GEN p33 = gen_0, p34 = gen_0; GEN p35 = gen_0; /* vec */ long l36; GEN p37 = gen_0; /* vec */ factD = gcopy(factD); if (gcmpgs(DEBUGLEVEL_qfsolve, 1) >= 0) pari_printf(" starting Qfsolve\n"); /* */ /* 1st reduction of the coefficients of G */ /* */ n = stoi(glength(G)); d = det(G); if (gequal0(d)) { p1 = gcopy(gel(matker0(G, 0), 1)); p1 = gerepilecopy(ltop, p1); return p1; } M = IndefiniteLLL(G, NULL, NULL, prec); if (typ(M) == t_COL) { if (gcmpgs(DEBUGLEVEL_qfsolve, 1) >= 0) pari_printf(" solution %Ps\n", M); if (gcmpgs(DEBUGLEVEL_qfsolve, 1) >= 0) pari_printf(" end of Qfsolve\n"); M = gerepilecopy(ltop, M); return M; } G = gcopy(gel(M, 1)); M = gcopy(gel(M, 2)); /* Real solubility */ signG = qfsign(G); if (gequal0(gel(signG, 1)) || gequal0(gel(signG, 2))) { if (gcmpgs(DEBUGLEVEL_qfsolve, 1) >= 0) pari_printf(" no real solution\n"); if (gcmpgs(DEBUGLEVEL_qfsolve, 1) >= 0) pari_printf(" end of Qfsolve\n"); avma = ltop; return gen_m1; } if (gcmp(gel(signG, 1), gel(signG, 2)) < 0) { G = gneg(G); p2 = cgetg(3, t_MAT); gel(p2, 1) = cgetg(3, t_COL); gel(p2, 2) = cgetg(3, t_COL); gcoeff(p2, 1, 1) = gen_0; gcoeff(p2, 1, 2) = gen_1; gcoeff(p2, 2, 1) = gen_1; gcoeff(p2, 2, 2) = gen_0; signG = gmul(signG, p2); } /* Factorization of the determinant */ if (gequal0(factD)) { if (gcmpgs(DEBUGLEVEL_qfsolve, 1) >= 0) pari_printf(" factorization of the determinant\n"); factD = factor(gneg(gabs(gmulsg(2, d), prec))); if (gcmpgs(DEBUGLEVEL_qfsolve, 1) >= 0) pari_printf("%Ps\n", factD); } gcoeff(factD, 1, 2) = gen_0; gcoeff(factD, 2, 2) = gsubgs(gcoeff(factD, 2, 2), 1); /* */ /* Minimization and local solubility */ /* */ if (gcmpgs(DEBUGLEVEL_qfsolve, 1) >= 0) pari_printf(" minimization of the determinant\n"); Min = Qfminim(G, factD, prec); if (typ(Min) == t_INT) { if (gcmpgs(DEBUGLEVEL_qfsolve, 1) >= 0) pari_printf(" no local solution at %Ps\n", Min); if (gcmpgs(DEBUGLEVEL_qfsolve, 1) >= 0) pari_printf(" end of Qfsolve\n"); Min = gerepilecopy(ltop, Min); return Min; } M = gmul(M, gel(Min, 2)); G = gcopy(gel(Min, 1)); /* Min[3] contains the factorization of abs(matdet(G)); */ if (gcmpgs(DEBUGLEVEL_qfsolve, 4) >= 0) pari_printf(" G minim = %Ps\n", G); if (gcmpgs(DEBUGLEVEL_qfsolve, 4) >= 0) pari_printf(" d = %Ps\n", d); /* Now, we know that local solutions exist */ /* (except maybe at 2 if n==4), */ /* if n==3, det(G) = +-1 */ /* if n==4, or n is odd, det(G) is squarefree. */ /* if n>=6, det(G) has all its valuations <=2. */ /* Reduction of G and search for trivial solutions. */ /* When det G=+-1, such trivial solutions always exist. */ if (gcmpgs(DEBUGLEVEL_qfsolve, 1) >= 0) pari_printf(" reduction\n"); U = IndefiniteLLL(G, NULL, NULL, prec); if (typ(U) == t_COL) { if (gcmpgs(DEBUGLEVEL_qfsolve, 1) >= 0) pari_printf(" solution = %Ps\n", gmul(M, U)); if (gcmpgs(DEBUGLEVEL_qfsolve, 1) >= 0) pari_printf(" end of Qfsolve\n"); p3 = gmul(M, U); p3 = gerepilecopy(ltop, p3); return p3; } G = gcopy(gel(U, 1)); M = gmul(M, gel(U, 2)); /* */ /* If n >= 6 is even, need to increment the dimension by 1 */ /* to suppress all the squares of det(G). */ /* */ if (((gcmpgs(n, 6) >= 0) && gequal0(gmodgs(n, 2))) && !gequalgs(gel(matsize(gel(Min, 3)), 1), 0)) { if (gcmpgs(DEBUGLEVEL_qfsolve, 1) >= 0) pari_printf(" increase the dimension by 1 = %Ps\n", gaddgs(n, 1)); codim = gen_1; n = gaddgs(n, 1); p4 = gcopy(gel(matsize(gel(Min, 3)), 1)); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN i = gen_0, p38 = gen_0; p5 = gen_1; for (i = gen_1; gcmp(i, p4) <= 0; i = gaddgs(i, 1)) { if (gequalgs(gcoeff(gel(Min, 3), gtos(i), 1), 2)) p38 = gcopy(gcoeff(gel(Min, 3), gtos(i), 1)); else p38 = gen_1; p5 = gmul(p5, p38); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 3, &p38, &p5, &i); } } /* largest square divisor of d. */ aux = p5; /* Choose the sign of aux such that the signature of G1 */ /* is as balanced as possible */ if (gcmp(gel(signG, 1), gel(signG, 2)) > 0) { gel(signG, 2) = gaddgs(gel(signG, 2), 1); aux = gneg(aux); } else gel(signG, 1) = gaddgs(gel(signG, 1), 1); p6 = cgetg(3, t_VEC); gel(p6, 1) = gcopy(G); gel(p6, 2) = gtomat(aux); G1 = matdiagonalblock(p6); detG1 = gmulsg(2, det(G1)); l7 = glength(gel(factD, 1)); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; for (i = 2; i <= l7; ++i) { gcoeff(factD, i, 2) = stoi(ggval(detG1, gcoeff(factD, i, 1))); if (low_stack(st_lim, stack_lim(btop, 1))) factD = gerepilecopy(btop, factD); } } gcoeff(factD, 2, 2) = gsubgs(gcoeff(factD, 2, 2), 1); Min = Qfminim(G1, factD, prec); G1 = gcopy(gel(Min, 1)); M1 = gcopy(gel(Min, 2)); p8 = gsubgs(n, 1); { long i, j; p9 = cgetg(gtos(p8)+1, t_MAT); for (j = 1; gcmpsg(j, p8) <= 0; ++j) { gel(p9, j) = cgetg(gtos(n)+1, t_COL); for (i = 1; gcmpsg(i, n) <= 0; ++i) gcoeff(p9, i, j) = stoi(i == j); } } subspace1 = p9; } else { codim = gen_0; G1 = gcopy(G); subspace1 = M1 = matid(gtos(n)); } /* Now, d is squarefree */ /* */ /* If d is not +-1, need to increment the dimension by 2 */ /* */ if (gequal0(gel(matsize(gel(Min, 3)), 1))) { /* if( abs(d) == 1, */ if (gcmpgs(DEBUGLEVEL_qfsolve, 2) >= 0) pari_printf(" detG2 = 1\n"); G2 = G1; subspace2 = M2 = matid(gtos(n)); solG2 = LLLgoon(G2, gen_1, prec); } else { if (gcmpgs(DEBUGLEVEL_qfsolve, 1) >= 0) pari_printf(" increase the dimension by 2 = %Ps\n", gaddgs(n, 2)); codim = gaddgs(codim, 2); p10 = gaddgs(n, 2); { long i, j; p11 = cgetg(gtos(n)+1, t_MAT); for (j = 1; gcmpsg(j, n) <= 0; ++j) { gel(p11, j) = cgetg(gtos(p10)+1, t_COL); for (i = 1; gcmpsg(i, p10) <= 0; ++i) gcoeff(p11, i, j) = stoi(i == j); } } subspace2 = p11; p12 = gcopy(gel(matsize(gel(Min, 3)), 1)); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN i = gen_0; p13 = gen_1; for (i = gen_1; gcmp(i, p12) <= 0; i = gaddgs(i, 1)) { p13 = gmul(p13, gcoeff(gel(Min, 3), gtos(i), 1)); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &p13, &i); } } d = p13; /* d = abs(matdet(G1)); */ if (gequal1(gmodgs(gel(signG, 2), 2))) d = gneg(d); /* d = matdet(G1); */ if (gequalgs(gcoeff(gel(Min, 3), 1, 1), 2)) { p14 = cgetg(2, t_VEC); gel(p14, 1) = gen_m1; factD = p14; } else { p15 = cgetg(3, t_VEC); gel(p15, 1) = gen_m1; gel(p15, 2) = gen_2; factD = p15; } /* if d is even ... */ factD = concat(factD, gtrans(gel(gel(Min, 3), 1))); if (gcmpgs(DEBUGLEVEL_qfsolve, 4) >= 0) pari_printf(" factD = %Ps\n", factD); /* Solubility at 2 (this is the only remaining bad prime). */ if (gequalgs(n, 4) && gequal1(gmodgs(d, 8))) if (gequal1(QfWittinvariant(G, gen_2))) { if (gcmpgs(DEBUGLEVEL_qfsolve, 1) >= 0) pari_printf(" no local solution at 2\n"); if (gcmpgs(DEBUGLEVEL_qfsolve, 1) >= 0) pari_printf(" end of Qfsolve\n"); avma = ltop; return gen_2; } l16 = glength(factD); { long l39; p17 = cgetg(l16+1, t_COL); for (l39 = 1; l39 <= l16; ++l39) gel(p17, l39) = gen_0; } /* */ /* Build a binary quadratic form with given invariants */ /* */ Winvariants = p17; /* choose the signature of Q. */ /* (real invariant and sign of the discriminant) */ dQ = gabs(d, prec); if (gequal(gel(signG, 1), gel(signG, 2))) { dQ = dQ; gel(Winvariants, 1) = gen_0; } /* signQ = [1,1]; */ if (gcmp(gel(signG, 1), gel(signG, 2)) > 0) { dQ = gneg(dQ); gel(Winvariants, 1) = gen_0; } /* signQ = [2,0]; */ if (gequalgs(n, 4) && !gequalgs(gmodgs(dQ, 4), 1)) dQ = gmulgs(dQ, 4); if (gcmpgs(n, 5) >= 0) dQ = gmulgs(dQ, 8); /* p-adic invariants */ /* for p = 2, the choice is fixed from the product formula */ if (gequalgs(n, 4)) { if (gcmpgs(DEBUGLEVEL_qfsolve, 1) >= 0) pari_printf(" compute the local invariants of G1\n"); aux = gcopy(gel(gel(Qflisteinvariants(gneg(G1), factD, prec), 2), 1)); l18 = glength(factD); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; for (i = 3; i <= l18; ++i) { gel(Winvariants, i) = gcopy(gel(aux, i)); if (low_stack(st_lim, stack_lim(btop, 1))) Winvariants = gerepilecopy(btop, Winvariants); } } } else { aux = gdiv(gmul(gpow(gen_m1, gdivgs(gsubgs(n, 3), 2), prec), dQ), d); /* ici aux = 8 ou -8 */ l19 = glength(factD); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; for (i = 3; i <= l19; ++i) { gel(Winvariants, i) = stoi(hilbert(aux, gel(factD, i), gel(factD, i)) > 0); if (low_stack(st_lim, stack_lim(btop, 1))) Winvariants = gerepilecopy(btop, Winvariants); } } } l20 = glength(factD); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; p21 = gen_0; for (i = 1; i <= l20; ++i) { p21 = gadd(p21, gel(Winvariants, i)); if (low_stack(st_lim, stack_lim(btop, 1))) p21 = gerepilecopy(btop, p21); } } gel(Winvariants, 2) = gmodgs(p21, 2); if (gcmpgs(DEBUGLEVEL_qfsolve, 1) >= 0) { pari_printf(" Search for a binary quadrativ form of discriminant = %Ps\n", dQ); pari_printf(" and Witt invariants = %Ps\n", Winvariants); } l22 = glength(factD) - 1; { long l40, l41; p23 = cgetg(3, t_MAT); for (l41 = 1; l41 <= 2; ++l41) { gel(p23, l41) = cgetg(l22+1, t_COL); for (l40 = 1; l40 <= l22; ++l40) gcoeff(p23, l40, l41) = gen_0; } } /* Construction of the 2-class group of discriminant dQ */ /* until some product of the generators gives the desired invariants. */ /* In dim 4, need to look among the form of the type q or 2*q */ /* because Q might be imprimitive. */ factd = p23; l24 = glength(factD) - 1; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN i = gen_0; for (i = gen_1; gcmpgs(i, l24) <= 0; i = gaddgs(i, 1)) { gcoeff(factd, gtos(i), 1) = gcopy(gel(factD, gtos(gaddgs(i, 1)))); gcoeff(factd, gtos(i), 2) = stoi(ggval(dQ, gcoeff(factd, gtos(i), 1))); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &i, &factd); } } gcoeff(factd, 1, 2) = gaddgs(gcoeff(factd, 1, 2), 1); l25 = gequalgs(n, 4); l26 = glength(factD); { long i, j; p27 = cgetg(l25+1, t_MAT); for (j = 1; j <= l25; ++j) { gel(p27, j) = cgetg(l26+1, t_COL); for (i = 1; i <= l26; ++i) gcoeff(p27, i, j) = stoi(hilbert(gen_2, dQ, gel(factD, i)) < 0); } } U2 = p27; clgp2 = class2(dQ, factd, Winvariants, U2, prec); if (gcmpgs(DEBUGLEVEL_qfsolve, 4) >= 0) pari_printf(" clgp2 = %Ps\n", clgp2); clgp2 = gcopy(gel(clgp2, 2)); U = gcopy(gel(Qflisteinvariants(clgp2, factD, prec), 2)); if (gequalgs(n, 4)) U = concat(U, U2); if (gcmpgs(DEBUGLEVEL_qfsolve, 4) >= 0) pari_printf(" U = %Ps\n", U); /* MODI2.1 */ V = lift(inverseimage(gmul(U, gmodulss(1, 2)), gmul(Winvariants, gmodulss(1, 2)))); /* if( !length(V), next); */ /* MODI 1 */ if (gcmpgs(DEBUGLEVEL_qfsolve, 4) >= 0) pari_printf(" V = %Ps\n", V); if (gequal1(gmodgs(dQ, 2))) Q = primeform(gmulsg(4, dQ), gen_1, prec); else Q = primeform(dQ, gen_1, prec); l28 = glength(clgp2); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; for (i = 1; i <= l28; ++i) { if (!gequal0(gel(V, i))) Q = qfbcompraw(Q, gel(clgp2, i)); if (low_stack(st_lim, stack_lim(btop, 1))) Q = gerepilecopy(btop, Q); } } Q = mymat(Q); if (gcmpgs(gnorml2(V), 1) > 0) { aux = QfbReduce(Q, prec); Q = gmul(gmul(gtrans(aux), Q), aux); } if (gequalgs(n, 4) && !gequal0(gel(V, glength(V)))) Q = gmulgs(Q, 2); if (gcmpgs(DEBUGLEVEL_qfsolve, 2) >= 0) pari_printf(" Q = %Ps\n", Q); if (gcmpgs(DEBUGLEVEL_qfsolve, 3) >= 0) pari_printf(" Witt invariants of Q = %Ps\n", Qflisteinvariants(Q, factD, prec)); p29 = cgetg(3, t_VEC); gel(p29, 1) = gcopy(G1); gel(p29, 2) = gneg(Q); /* */ /* Build a form of dim=n+2 potentially unimodular */ /* */ G2 = matdiagonalblock(p29); if (gcmpgs(DEBUGLEVEL_qfsolve, 4) >= 0) pari_printf(" G2 = %Ps\n", G2); if (gcmpgs(DEBUGLEVEL_qfsolve, 2) >= 0) pari_printf(" minimization of the form of dimension %ld\n", glength(G2)); /* Minimization of G2 */ detG2 = det(G2); l30 = glength(factD) - 1; { long l42, l43; p31 = cgetg(3, t_MAT); for (l43 = 1; l43 <= 2; ++l43) { gel(p31, l43) = cgetg(l30+1, t_COL); for (l42 = 1; l42 <= l30; ++l42) gcoeff(p31, l42, l43) = gen_0; } } factd = p31; l32 = glength(factD) - 1; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN i = gen_0; for (i = gen_1; gcmpgs(i, l32) <= 0; i = gaddgs(i, 1)) { gcoeff(factd, gtos(i), 2) = stoi(ggval(detG2, gcoeff(factd, gtos(i), 1) = gcopy(gel(factD, gtos(gaddgs(i, 1)))))); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &i, &factd); } } if (gcmpgs(DEBUGLEVEL_qfsolve, 3) >= 0) pari_printf(" det(G2) = %Ps\n", factd); Min = Qfminim(G2, factd, prec); M2 = gcopy(gel(Min, 2)); G2 = gcopy(gel(Min, 1)); if (gcmpgs(gabs(det(G2), prec), 2) > 0) pari_err(user, "Qfsolve: det(G2) <> +-1 *******"); if (gcmpgs(DEBUGLEVEL_qfsolve, 4) >= 0) pari_printf(" G2 = %Ps\n", G2); /* Now, we have det(G2) = +-1 */ /* Find a seti for G2 (Totally isotropic subspace, Sous-Espace Totalement Isotrope) */ if (gcmpgs(DEBUGLEVEL_qfsolve, 2) >= 0) pari_printf(" Search a subspace of solutions for G2\n"); solG2 = LLLgoon(G2, gen_1, prec); p33 = gaddgs(codim, 1); p34 = gaddgs(codim, 1); { long i, j; p35 = cgetg(gtos(p33)+1, t_MAT); for (j = 1; gcmpsg(j, p33) <= 0; ++j) { gel(p35, j) = cgetg(gtos(p34)+1, t_COL); for (i = 1; gcmpsg(i, p34) <= 0; ++i) gcoeff(p35, i, j) = gcopy(gcoeff(gel(solG2, 1), i, j)); } } if (!gequalgs(p35, 0)) pari_err(user, "Qfsolve: not enough solutions in G2"); } /* G2 must have a subspace of solutions of dimension > codim */ dimseti = gen_0; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN p44 = gen_0, p45 = gen_0; GEN p46 = gen_0; /* vec */ for(;;) { p44 = gaddgs(dimseti, 1); p45 = gaddgs(dimseti, 1); { long i, j; p46 = cgetg(gtos(p44)+1, t_MAT); for (j = 1; gcmpsg(j, p44) <= 0; ++j) { gel(p46, j) = cgetg(gtos(p45)+1, t_COL); for (i = 1; gcmpsg(i, p45) <= 0; ++i) gcoeff(p46, i, j) = gcopy(gcoeff(gel(solG2, 1), i, j)); } } if (!gequal0(p46)) break; dimseti = gaddgs(dimseti, 1); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 4, &p44, &p45, &p46, &dimseti); } } if (gcmp(dimseti, codim) <= 0) pari_err(user, "Qfsolve: not enough solutions for G2"); l36 = glength(G2); { long i, j; p37 = cgetg(gtos(dimseti)+1, t_MAT); for (j = 1; gcmpsg(j, dimseti) <= 0; ++j) { gel(p37, j) = cgetg(l36+1, t_COL); for (i = 1; i <= l36; ++i) gcoeff(p37, i, j) = gcopy(gcoeff(gel(solG2, 2), i, j)); } } solG2 = p37; if (gcmpgs(DEBUGLEVEL_qfsolve, 3) >= 0) pari_printf(" solG2 = %Ps\n", solG2); /* The solution of G1 is simultaneously in solG2 and subspace2 */ if (gcmpgs(DEBUGLEVEL_qfsolve, 1) >= 0) pari_printf(" Reconstruction of a solution of G1\n"); solG1 = intersect(subspace2, gmul(M2, solG2)); solG1 = gmul(gtrans(subspace2), solG1); if (gcmpgs(DEBUGLEVEL_qfsolve, 3) >= 0) pari_printf(" solG1 = %Ps\n", solG1); /* The solution of G is simultaneously in solG and subspace1 */ if (gcmpgs(DEBUGLEVEL_qfsolve, 1) >= 0) pari_printf(" Reconstruction of a solution of G\n"); sol = intersect(subspace1, gmul(M1, solG1)); sol = gmul(gtrans(subspace1), sol); sol = gmul(M, sol); sol = gdiv(sol, content(sol)); if (glength(sol) == 1) sol = gcopy(gel(sol, 1)); if (gcmpgs(DEBUGLEVEL_qfsolve, 3) >= 0) pari_printf(" sol = %Ps\n", sol); if (gcmpgs(DEBUGLEVEL_qfsolve, 1) >= 0) pari_printf(" end of Qfsolve\n"); sol = gerepilecopy(ltop, sol); return sol; } GEN matdiagonalblock(GEN v) { pari_sp ltop = avma; GEN lv = gen_0, lt = gen_0, M = gen_0, p1 = gen_0; GEN p2 = gen_0; /* vec */ lv = stoi(glength(v)); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN i = gen_0; p1 = gen_0; for (i = gen_1; gcmp(i, lv) <= 0; i = gaddgs(i, 1)) { p1 = gaddgs(p1, glength(gel(v, gtos(i)))); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &p1, &i); } } lt = p1; { long l3, l4; p2 = cgetg(gtos(lt)+1, t_MAT); for (l4 = 1; gcmpsg(l4, lt) <= 0; ++l4) { gel(p2, l4) = cgetg(gtos(lt)+1, t_COL); for (l3 = 1; gcmpsg(l3, lt) <= 0; ++l3) gcoeff(p2, l3, l4) = gen_0; } } M = p2; lt = gen_0; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN i = gen_0; long l5; for (i = gen_1; gcmp(i, lv) <= 0; i = gaddgs(i, 1)) { l5 = glength(gel(v, gtos(i))); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long j, l6; for (j = 1; j <= l5; ++j) { l6 = glength(gel(v, gtos(i))); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long k; for (k = 1; k <= l6; ++k) { gcoeff(M, gtos(gaddgs(lt, j)), gtos(gaddgs(lt, k))) = gcopy(gcoeff(gel(v, gtos(i)), j, k)); if (low_stack(st_lim, stack_lim(btop, 1))) M = gerepilecopy(btop, M); } } if (low_stack(st_lim, stack_lim(btop, 1))) M = gerepilecopy(btop, M); } } lt = gaddgs(lt, glength(gel(v, gtos(i)))); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 3, &i, &M, <); } } M = gerepilecopy(ltop, M); return M; } GEN ellchangecurveinverse(GEN ell, GEN v) /* ell */ { pari_sp ltop = avma; GEN p1 = gen_0; /* ell */ p1 = ellchangecurve(ell, ellinverturst(v)); p1 = gerepilecopy(ltop, p1); return p1; } GEN ellchangepointinverse(GEN pt, GEN v) { pari_sp ltop = avma; GEN p1 = gen_0; p1 = ellchangepoint(pt, ellinverturst(v)); p1 = gerepilecopy(ltop, p1); return p1; } GEN ellcomposeurst(GEN urst1, GEN urst2) /* vec */ { pari_sp ltop = avma; GEN u1 = gen_0, r1 = gen_0, s1 = gen_0, t1 = gen_0, u2 = gen_0, r2 = gen_0, s2 = gen_0, t2 = gen_0; GEN p1 = gen_0; /* vec */ t2 = gcopy(gel(urst2, 4)); s2 = gcopy(gel(urst2, 3)); r2 = gcopy(gel(urst2, 2)); u2 = gcopy(gel(urst2, 1)); t1 = gcopy(gel(urst1, 4)); s1 = gcopy(gel(urst1, 3)); r1 = gcopy(gel(urst1, 2)); u1 = gcopy(gel(urst1, 1)); p1 = cgetg(5, t_VEC); gel(p1, 1) = gmul(u1, u2); gel(p1, 2) = gadd(gmul(gsqr(u1), r2), r1); gel(p1, 3) = gadd(gmul(u1, s2), s1); gel(p1, 4) = gadd(gadd(gmul(gpowgs(u1, 3), t2), gmul(gmul(s1, gsqr(u1)), r2)), t1); p1 = gerepilecopy(ltop, p1); return p1; } GEN ellinverturst(GEN urst) /* vec */ { pari_sp ltop = avma; GEN u = gen_0, r = gen_0, s = gen_0, t = gen_0; GEN p1 = gen_0; /* vec */ t = gcopy(gel(urst, 4)); s = gcopy(gel(urst, 3)); r = gcopy(gel(urst, 2)); u = gcopy(gel(urst, 1)); p1 = cgetg(5, t_VEC); gel(p1, 1) = ginv(u); gel(p1, 2) = gdiv(gneg(r), gsqr(u)); gel(p1, 3) = gdiv(gneg(s), u); gel(p1, 4) = gdiv(gsub(gmul(r, s), t), gpowgs(u, 3)); p1 = gerepilecopy(ltop, p1); return p1; } GEN mysubst(GEN polsu, GEN subsx) { pari_sp ltop = avma; GEN p1 = gen_0, p2 = gen_0; if (typ(lift(polsu)) == t_POL) { p1 = simplify(gsubst(lift(polsu), gvar(gpolvar(lift(polsu))), subsx)); p1 = gerepilecopy(ltop, p1); return p1; } else { p2 = simplify(lift(polsu)); p2 = gerepilecopy(ltop, p2); return p2; } avma = ltop; return gen_0; } GEN degre(GEN idegre) { pari_sp ltop = avma; GEN ideg = gen_0, jdeg = gen_0; ideg = gcopy(idegre); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); while (signe(ideg = gshift(ideg, -1))) { jdeg = gaddgs(jdeg, 1); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &ideg, &jdeg); } } jdeg = gerepilecopy(ltop, jdeg); return jdeg; } long nfissquare(GEN nf, GEN a, long prec) /* bool */ { pari_sp ltop = avma; long l1; /* bool */ l1 = (lg(nfsqrt(nf, a, prec))-1) > 0; avma = ltop; return l1; } GEN nfsqrt(GEN nf, GEN a, long prec) /* vec */ { pari_sp ltop = avma; GEN alift = gen_0, ta = gen_0, py = gen_0, pfact = gen_0; GEN p1 = gen_0, p2 = gen_0, p3 = gen_0; /* vec */ GEN p4 = gen_0; /* int */ GEN x = pol_x(fetch_user_var("x")), y = pol_x(fetch_user_var("y")); GEN p5 = gen_0, p6 = gen_0; /* vec */ if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" starting nfsqrt %Ps\n", a); if (gequal0(a) || gequal1(a)) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" end of nfsqrt %Ps\n", a); p1 = cgetg(2, t_VEC); gel(p1, 1) = gcopy(a); p1 = gerepilecopy(ltop, p1); return p1; } alift = lift(a); ta = strtoGENstr(type_name(typ(a))); if (!degree(alift)) alift = polcoeff0(alift, 0, -1); if (typ(alift) != t_POL) if (!gequal0(gissquare(alift))) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" end of nfsqrt %Ps\n", sqrtrat(alift)); p2 = cgetg(2, t_VEC); gel(p2, 1) = sqrtrat(alift); p2 = gerepilecopy(ltop, p2); return p2; } if (degree(member_pol(nf)) <= 1) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" end of nfsqrt %Ps\n", cgetg(1, t_VEC)); p3 = cgetg(1, t_VEC); p3 = gerepilecopy(ltop, p3); return p3; } if (gequal(ta, strtoGENstr("t_POL"))) a = gmodulo(a, member_pol(nf)); /* the real embeddings must all be >0 */ p4 = icopy(member_r1(nf)); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN i = gen_0; GEN p7 = gen_0; /* vec */ for (i = gen_1; gcmp(i, p4) <= 0; i = gaddgs(i, 1)) { /* MODI: in ell.gp, nfsign_s(nf,a,i) < 0 so replaced. Both correct or only one ? */ /* py = mysubst(alift,nf.roots[i]); if( sign(py) < 0, */ if (nfsign_s(nf, a, i, prec) < 0) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" end of nfsqrt %Ps\n", cgetg(1, t_VEC)); p7 = cgetg(1, t_VEC); p7 = gerepilecopy(ltop, p7); return p7; } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &i, &p7); } } /* factorization over nf of the polynomial X^2-a */ if (gequal(gpolvar(member_pol(nf)), x)) { py = gsubst(member_pol(nf), gvar(x), y); pfact = lift(gcoeff(polfnf(gsub(gsqr(x), mysubst(alift, gmodulo(y, py))), py), 1, 1)); } else pfact = lift(gcoeff(polfnf(gsub(gsqr(x), a), member_pol(nf)), 1, 1)); if (degree(pfact) == 2) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" end of nfsqrt %Ps\n", cgetg(1, t_VEC)); p5 = cgetg(1, t_VEC); p5 = gerepilecopy(ltop, p5); return p5; } if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" end of nfsqrt %Ps\n", pfact); p6 = cgetg(2, t_VEC); gel(p6, 1) = gsubst(polcoeff0(pfact, 0, -1), gvar(y), gmodulo(gpolvar(member_pol(nf)), member_pol(nf))); p6 = gerepilecopy(ltop, p6); return p6; } GEN sqrtrat(GEN a) { pari_sp ltop = avma; GEN p1 = gen_0; p1 = gdiv(sqrtint(numer(a)), sqrtint(denom(a))); p1 = gerepilecopy(ltop, p1); return p1; } GEN polratroots(GEN pol) { pari_sp ltop = avma; GEN f = gen_0, ans = gen_0; long l1; f = gcopy(gel(factor(pol), 1)); ans = cgetg(1, t_VEC); l1 = glength(f); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long j; GEN p2 = gen_0; /* vec */ for (j = 1; j <= l1; ++j) { if (degree(gel(f, j)) == 1) { p2 = cgetg(2, t_VEC); gel(p2, 1) = gdiv(gneg(polcoeff0(gel(f, j), 0, -1)), polcoeff0(gel(f, j), 1, -1)); ans = concat(ans, p2); } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &p2, &ans); } } ans = gerepilecopy(ltop, ans); return ans; } GEN ratpoint(GEN pol, GEN lim, GEN singlepoint, long prec) { pari_sp ltop = avma; GEN listpoints = gen_0, point1 = gen_0, odd = gen_0, deg4 = gen_0, pol16 = gen_0, tab16 = gen_0, pol9 = gen_0, tab9 = gen_0, pol5 = gen_0, tab5 = gen_0, pol0 = gen_0, vecz = gen_0, vecx = gen_0, lead = gen_0, zz = gen_0, xx = gen_0, evpol = gen_0, iz = gen_0, factpol = gen_0, deg = gen_0, vz = gen_0; GEN p1 = gen_0, p2 = gen_0, p3 = gen_0, p4 = gen_0, p5 = gen_0; /* vec */ long l6; GEN p7 = gen_0, p8 = gen_0; /* vec */ long l9; GEN p10 = gen_0; /* vec */ long l11; GEN p12 = gen_0; /* vec */ long l13; GEN p14 = gen_0, p15 = gen_0, p16 = gen_0, p17 = gen_0; /* vec */ GEN p18 = gen_0, x = pol_x(fetch_user_var("x")); GEN p19 = gen_0; /* vec */ if (!lim) lim = gen_1; if (!singlepoint) singlepoint = gen_1; /* MODI: added deg, vz */ if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) { pari_printf(" Starting ratpoint with pol = %Ps\n", pol); pari_printf(" lim = %Ps\n", lim); } if (gequal0(singlepoint)) listpoints = cgetg(1, t_VEC); point1 = cgetg(1, t_VEC); /* */ /* trivial solutions */ /* */ /* the leading coeff is a square */ if (!gequal0(gissquare(pollead(pol, -1)))) { p1 = cgetg(4, t_VEC); gel(p1, 1) = gen_1; gel(p1, 2) = sqrtrat(pollead(pol, -1)); gel(p1, 3) = gen_0; point1 = p1; if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" trivial solution: lead(pol) is a square\n"); if (!gequal0(singlepoint)) { if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" end of ratpoint\n"); point1 = gerepilecopy(ltop, point1); return point1; } p2 = cgetg(2, t_VEC); gel(p2, 1) = gcopy(point1); listpoints = concat(listpoints, p2); } /* the constant coeff is a square */ if (!gequal0(gissquare(polcoeff0(pol, 0, -1)))) { p3 = cgetg(3, t_VEC); gel(p3, 1) = gen_0; gel(p3, 2) = sqrtrat(polcoeff0(pol, 0, -1)); point1 = p3; if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" trivial solution: pol(0) is a square\n"); if (!gequal0(singlepoint)) { if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" end of ratpoint\n"); point1 = gerepilecopy(ltop, point1); return point1; } p4 = cgetg(2, t_VEC); gel(p4, 1) = gcopy(point1); listpoints = concat(listpoints, p4); } odd = stoi(smodss(degree(pol), 2)); /* roots of pol ? */ factpol = polratroots(pol); if (glength(factpol)) { if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" trivial solution: roots of pol%Ps\n", factpol); if (!gequal0(singlepoint)) { if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" end of ratpoint\n"); p5 = cgetg(3, t_VEC); gel(p5, 1) = gcopy(gel(factpol, 1)); gel(p5, 2) = gen_0; p5 = gerepilecopy(ltop, p5); return p5; } l6 = glength(factpol); { long i; GEN p20 = gen_0; /* vec */ p7 = cgetg(l6+1, t_VEC); for (i = 1; i <= l6; ++i) { p20 = cgetg(3, t_VEC); gel(p20, 1) = gcopy(gel(factpol, i)); gel(p20, 2) = gen_0; gel(p7, i) = p20; } } listpoints = concat(listpoints, p7); } /* */ /* Sieve */ /* */ /* initialisation of the sieve modulo 16, 9 and 5 */ /* used only with even degree when lim is large */ deg = stoi(degree(pol)); deg4 = stoi((gequal0(odd)) && (gcmpgs(lim, 20) > 0)); if (!gequal0(deg4)) { pol16 = gtrans(gmul(gtovec(pol), gmodulss(1, 16))); { long l21, l22; p8 = cgetg(17, t_MAT); for (l22 = 1; l22 <= 16; ++l22) { gel(p8, l22) = cgetg(17, t_COL); for (l21 = 1; l21 <= 16; ++l21) gcoeff(p8, l21, l22) = gen_0; } } tab16 = p8; l9 = 16 - 1; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN xx = gen_0; long l23; for (xx = gen_0; gcmpgs(xx, l9) <= 0; xx = gaddgs(xx, 1)) { l23 = 16 - 1; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN zz = gen_0, p24 = gen_0; GEN p25 = gen_0; /* vec */ for (zz = gen_0; gcmpgs(zz, l23) <= 0; zz = gaddgs(zz, 1)) { p24 = gaddgs(deg, 1); { long i; p25 = cgetg(gtos(p24)+1, t_VEC); for (i = 1; gcmpsg(i, p24) <= 0; ++i) gel(p25, i) = gmul(gpow(xx, gsubgs(gaddgs(deg, 1), i), prec), gpowgs(zz, i - 1)); } gcoeff(tab16, gtos(gaddgs(xx, 1)), gtos(gaddgs(zz, 1))) = stoi(gequal0(gissquare(gmul(p25, pol16)))); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 4, &zz, &p24, &p25, &tab16); } } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &xx, &tab16); } } pol9 = gmul(gtrans(gtovec(pol)), gmodulss(1, 9)); { long l26, l27; p10 = cgetg(10, t_MAT); for (l27 = 1; l27 <= 9; ++l27) { gel(p10, l27) = cgetg(10, t_COL); for (l26 = 1; l26 <= 9; ++l26) gcoeff(p10, l26, l27) = gen_0; } } tab9 = p10; l11 = 9 - 1; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN xx = gen_0; long l28; for (xx = gen_0; gcmpgs(xx, l11) <= 0; xx = gaddgs(xx, 1)) { l28 = 9 - 1; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN zz = gen_0, p29 = gen_0; GEN p30 = gen_0; /* vec */ for (zz = gen_0; gcmpgs(zz, l28) <= 0; zz = gaddgs(zz, 1)) { p29 = gaddgs(deg, 1); { long i; p30 = cgetg(gtos(p29)+1, t_VEC); for (i = 1; gcmpsg(i, p29) <= 0; ++i) gel(p30, i) = gmul(gpow(xx, gsubgs(gaddgs(deg, 1), i), prec), gpowgs(zz, i - 1)); } gcoeff(tab9, gtos(gaddgs(xx, 1)), gtos(gaddgs(zz, 1))) = stoi(gequal0(gissquare(gmul(p30, pol9)))); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 4, &zz, &p29, &p30, &tab9); } } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &xx, &tab9); } } pol5 = gmul(gtrans(gtovec(pol)), gmodulss(1, 5)); { long l31, l32; p12 = cgetg(6, t_MAT); for (l32 = 1; l32 <= 5; ++l32) { gel(p12, l32) = cgetg(6, t_COL); for (l31 = 1; l31 <= 5; ++l31) gcoeff(p12, l31, l32) = gen_0; } } tab5 = p12; l13 = 5 - 1; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN xx = gen_0; long l33; for (xx = gen_0; gcmpgs(xx, l13) <= 0; xx = gaddgs(xx, 1)) { l33 = 5 - 1; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN zz = gen_0, p34 = gen_0; GEN p35 = gen_0; /* vec */ for (zz = gen_0; gcmpgs(zz, l33) <= 0; zz = gaddgs(zz, 1)) { p34 = gaddgs(deg, 1); { long i; p35 = cgetg(gtos(p34)+1, t_VEC); for (i = 1; gcmpsg(i, p34) <= 0; ++i) gel(p35, i) = gmul(gpow(xx, gsubgs(gaddgs(deg, 1), i), prec), gpowgs(zz, i - 1)); } gcoeff(tab5, gtos(gaddgs(xx, 1)), gtos(gaddgs(zz, 1))) = stoi(gequal0(gissquare(gmul(p35, pol5)))); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 4, &zz, &p34, &p35, &tab5); } } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &xx, &tab5); } } } /* if the degree is odd, search only for square denominators */ if (!gequal0(odd)) { { long i; p14 = cgetg(gtos(lim)+1, t_VEC); for (i = 1; gcmpsg(i, lim) <= 0; ++i) gel(p14, i) = sqri(stoi(i)); } vecz = p14; } else { /* if the degree is even, the leading coeff must be */ /* a square modulo zz. */ lead = pollead(pol, -1); { long l36; p15 = cgetg(gtos(lim)+1, t_VEC); for (l36 = 1; gcmpsg(l36, lim) <= 0; ++l36) gel(p15, l36) = gen_0; } vecz = p15; zz = gen_0; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN i = gen_0; for (i = gen_1; gcmp(i, lim) <= 0; i = gaddgs(i, 1)) { zz = gaddgs(zz, 1); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); while (gequal0(gissquare(gmodulo(lead, zz)))) { zz = gaddgs(zz, 1); if (low_stack(st_lim, stack_lim(btop, 1))) zz = gerepilecopy(btop, zz); } } gel(vecz, gtos(i)) = gcopy(zz); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 3, &i, &zz, &vecz); } } } /* the constant coeff must be a square modulo xx. */ pol0 = polcoeff0(pol, 0, -1); { long l37; p16 = cgetg(gtos(lim)+1, t_VEC); for (l37 = 1; gcmpsg(l37, lim) <= 0; ++l37) gel(p16, l37) = gen_0; } vecx = p16; xx = gen_0; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN i = gen_0; for (i = gen_1; gcmp(i, lim) <= 0; i = gaddgs(i, 1)) { xx = gaddgs(xx, 1); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); while (gequal0(gissquare(gmodulo(pol0, xx)))) { xx = gaddgs(xx, 1); if (low_stack(st_lim, stack_lim(btop, 1))) xx = gerepilecopy(btop, xx); } } gel(vecx, gtos(i)) = gcopy(xx); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 3, &i, &xx, &vecx); } } if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" xmax = %Ps\n", gel(vecx, gtos(lim))); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" zmax = %Ps\n", gel(vecz, gtos(lim))); if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" vecx = %Ps\n", vecx); if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" vecz = %Ps\n", vecz); if (!gequal0(deg4)) { { long i; GEN p38 = gen_0; GEN p39 = gen_0; /* vec */ p17 = cgetg(gtos(lim)+1, t_VEC); for (i = 1; gcmpsg(i, lim) <= 0; ++i) { p38 = gaddgs(deg, 1); { long j; p39 = cgetg(gtos(p38)+1, t_VEC); for (j = 1; gcmpsg(j, p38) <= 0; ++j) gel(p39, j) = gmul(polcoeff0(pol, gtos(gsubgs(gaddgs(deg, 1), j)), -1), gpowgs(gel(vecz, i), j - 1)); } gel(p17, i) = gtopoly(p39, -1); } } vz = p17; } /* loop over x = xx/zz */ /* the loop on [xx,zz] is done diagonally */ /* to start with the smallest values of both xx and zz. */ p18 = gmulsg(2, lim); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN somme = gen_0, p40 = gen_0, p41 = gen_0; for (somme = gen_2; gcmp(somme, p18) <= 0; somme = gaddgs(somme, 1)) { p40 = gmaxsg(1, gsub(somme, lim)); p41 = gmin(lim, gsubgs(somme, 1)); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN ix = gen_0; for (ix = p40; gcmp(ix, p41) <= 0; ix = gaddgs(ix, 1)) { xx = gcopy(gel(vecx, gtos(ix))); iz = gsub(somme, ix); zz = gcopy(gel(vecz, gtos(iz))); if (!gequalgs(ggcd(zz, xx), 1)) continue; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long eps; GEN p42 = gen_0, p43 = gen_0; /* vec */ for (eps = 1; eps <= 2; ++eps) { /* when eps = 1, xx > 0; when eps = 2, xx < 0. */ if (!gequal0(deg4)) { if (!gequal0(gcoeff(tab16, gtos(gaddgs(gmodgs(xx, 16), 1)), gtos(gaddgs(gmodgs(zz, 16), 1))))) { xx = gneg(xx); continue; } if (!gequal0(gcoeff(tab9, gtos(gaddgs(gmodgs(xx, 9), 1)), gtos(gaddgs(gmodgs(zz, 9), 1))))) { xx = gneg(xx); continue; } if (!gequal0(gcoeff(tab5, gtos(gaddgs(gmodgs(xx, 5), 1)), gtos(gaddgs(gmodgs(zz, 5), 1))))) { xx = gneg(xx); continue; } evpol = gsubst(gel(vz, gtos(iz)), gvar(x), xx); } else evpol = gsubst(pol, gvar(gpolvar(pol)), gdiv(xx, zz)); if (!gequal0(gissquare(evpol))) { p42 = cgetg(3, t_VEC); gel(p42, 1) = gdiv(xx, zz); gel(p42, 2) = sqrtrat(gsubst(pol, gvar(gpolvar(pol)), gdiv(xx, zz))); point1 = p42; if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" point found by ratpoint = %Ps\n", point1); if (!gequal0(singlepoint)) goto label1; p43 = cgetg(2, t_VEC); gel(p43, 1) = gcopy(point1); listpoints = concat(listpoints, p43); } xx = gneg(xx); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 6, &xx, &evpol, &p42, &point1, &p43, &listpoints); } } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 7, &ix, &xx, &iz, &zz, &evpol, &point1, &listpoints); } } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 9, &somme, &p40, &p41, &xx, &iz, &zz, &evpol, &point1, &listpoints); } label1:; } if (!gequal(point1, cgetg(1, t_VEC))) { if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" point found by ratpoint = %Ps\n", point1); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" end of ratpoint \n"); if (!gequal0(singlepoint)) { point1 = gerepilecopy(ltop, point1); return point1; } else { listpoints = gerepilecopy(ltop, listpoints); return listpoints; } } p19 = cgetg(1, t_VEC); p19 = gerepilecopy(ltop, p19); return p19; } GEN ratpoint2(GEN pol, GEN lim, GEN singlepoint, GEN redflag, long prec) { pari_sp ltop = avma; GEN listpoints = gen_0, list = gen_0, rr = gen_0, y2 = gen_0, aux = gen_0; long l1; if (!lim) lim = gen_1; if (!singlepoint) singlepoint = gen_1; if (!redflag) redflag = gen_0; listpoints = cgetg(1, t_VEC); list = listratpoint(pol, redflag, prec); l1 = glength(list); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; GEN p2 = gen_0; /* vec */ long l3; GEN p4 = gen_0; for (i = 1; i <= l1; ++i) { rr = ratpoint(gel(gel(list, i), 1), lim, singlepoint, prec); if (!gequal0(singlepoint) && glength(rr)) { p2 = cgetg(2, t_VEC); gel(p2, 1) = gcopy(rr); rr = p2; } l3 = glength(rr); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long j; GEN p5 = gen_0, p6 = gen_0, p7 = gen_0, p8 = gen_0; /* vec */ for (j = 1; j <= l3; ++j) { y2 = gmul(gel(gel(rr, j), 2), gel(gel(list, i), 4)); if (glength(gel(rr, j)) == 2) { p5 = cgetg(3, t_COL); gel(p5, 1) = gcopy(gel(gel(rr, j), 1)); gel(p5, 2) = gen_1; aux = p5; } else { p6 = cgetg(3, t_COL); gel(p6, 1) = gcopy(gel(gel(rr, j), 1)); gel(p6, 2) = gcopy(gel(gel(rr, j), 3)); aux = p6; } aux = gmul(gel(gel(list, i), 2), aux); if (gequal0(gel(aux, 2))) { p7 = cgetg(4, t_VEC); gel(p7, 1) = gcopy(gel(aux, 1)); gel(p7, 2) = gcopy(y2); gel(p7, 3) = gen_0; gel(rr, j) = p7; } else { p8 = cgetg(3, t_VEC); gel(p8, 1) = gdiv(gel(aux, 1), gel(aux, 2)); gel(p8, 2) = gdiv(y2, gpowgs(gel(aux, 2), degree(pol)/2)); gel(rr, j) = p8; } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 7, &y2, &p5, &aux, &p6, &p7, &rr, &p8); } } if (!gequal0(singlepoint) && glength(rr)) { p4 = gcopy(gel(rr, 1)); p4 = gerepilecopy(ltop, p4); return p4; } listpoints = concat(listpoints, rr); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 6, &rr, &p2, &y2, &aux, &p4, &listpoints); } } listpoints = vecsort0(listpoints, NULL, 2); listpoints = gerepilecopy(ltop, listpoints); return listpoints; } GEN listratpoint(GEN pol, GEN redflag, long prec) { pari_sp ltop = avma; GEN list = gen_0, i = gen_0, K = gen_0, ff = gen_0, C = gen_0, p = gen_0, M = gen_0, U = gen_0, newpol = gen_0, factpol = gen_0, ll = gen_0, listf = gen_0, rr = gen_0; GEN p1 = gen_0, p2 = gen_0; /* vec */ GEN x = pol_x(fetch_user_var("x")); long l3; GEN p4 = gen_0; GEN p5 = gen_0; /* vec */ long l6, l7; if (!redflag) redflag = gen_0; if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" Starting listratpoint with %Ps\n", pol); p1 = cgetg(2, t_VEC); p2 = cgetg(5, t_VEC); gel(p2, 1) = gcopy(pol); gel(p2, 2) = matid(2); gel(p2, 3) = gen_1; gel(p2, 4) = gen_1; gel(p1, 1) = p2; list = p1; i = gen_1; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN p8 = gen_0, p9 = gen_0, p10 = gen_0; /* vec */ long l11; while (gcmpgs(i, glength(list)) <= 0) { pol = gcopy(gel(gel(list, gtos(i)), 1)); K = gabs(content(pol), prec); if (!gequalgs(K, 1)) { pol = gel(gel(list, gtos(i)), 1) = gdiv(gel(gel(list, gtos(i)), 1), K); gel(gel(list, gtos(i)), 3) = gmul(gel(gel(list, gtos(i)), 3), K); } K = gcopy(gel(gel(list, gtos(i)), 3)); if (gequal1(K)) { i = gaddgs(i, 1); continue; } ff = factor(K); if (gcmpgs(vecmax(gel(ff, 2)), 1) > 0) { gel(ff, 2) = gdivent(gel(ff, 2), gen_2); C = factorback(ff); gel(gel(list, gtos(i)), 4) = gmul(gel(gel(list, gtos(i)), 4), C); K = gel(gel(list, gtos(i)), 3) = gdiv(gel(gel(list, gtos(i)), 3), gsqr(C)); if (gequal1(K)) { i = gaddgs(i, 1); continue; } ff = factor(K); } p = gcopy(gcoeff(ff, 1, 1)); M = gcopy(gel(gel(list, gtos(i)), 2)); C = gcopy(gel(gel(list, gtos(i)), 4)); if (gequal0(gmod(pollead(pol, -1), p))) { p8 = cgetg(3, t_MAT); gel(p8, 1) = cgetg(3, t_COL); gel(p8, 2) = cgetg(3, t_COL); gcoeff(p8, 1, 1) = gen_1; gcoeff(p8, 1, 2) = gen_0; gcoeff(p8, 2, 1) = gen_0; gcoeff(p8, 2, 2) = gcopy(p); U = gmul(M, p8); if (gequal1(content(U))) { newpol = gmul(gsubst(pol, gvar(x), gdiv(x, p)), gpowgs(p, degree(pol) - 1)); p9 = cgetg(2, t_VEC); p10 = cgetg(5, t_VEC); gel(p10, 1) = gcopy(newpol); gel(p10, 2) = gcopy(U); gel(p10, 3) = gdiv(K, p); gel(p10, 4) = gmul(C, p); gel(p9, 1) = p10; list = concat(list, p9); } } factpol = centerlift(rootmod(pol, p)); l11 = glength(factpol); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long j; GEN p12 = gen_0, p13 = gen_0, p14 = gen_0; /* vec */ for (j = 1; j <= l11; ++j) { p12 = cgetg(3, t_MAT); gel(p12, 1) = cgetg(3, t_COL); gel(p12, 2) = cgetg(3, t_COL); gcoeff(p12, 1, 1) = gcopy(p); gcoeff(p12, 1, 2) = gcopy(gel(factpol, j)); gcoeff(p12, 2, 1) = gen_0; gcoeff(p12, 2, 2) = gen_1; U = gmul(M, p12); if (gequal1(content(U))) { newpol = gdiv(gsubst(pol, gvar(x), gadd(gmul(p, x), gel(factpol, j))), p); p13 = cgetg(2, t_VEC); p14 = cgetg(5, t_VEC); gel(p14, 1) = gcopy(newpol); gel(p14, 2) = gcopy(U); gel(p14, 3) = gdiv(K, p); gel(p14, 4) = gmul(C, p); gel(p13, 1) = p14; list = concat(list, p13); } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 6, &p12, &U, &newpol, &p14, &p13, &list); } } i = gaddgs(i, 1); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 14, &pol, &K, &list, &i, &ff, &C, &p, &M, &p8, &U, &newpol, &p10, &p9, &factpol); } } l3 = glength(list); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; p4 = gen_0; for (i = 1; i <= l3; ++i) { p4 = gaddgs(p4, gequal1(gel(gel(list, i), 3))); if (low_stack(st_lim, stack_lim(btop, 1))) p4 = gerepilecopy(btop, p4); } } ll = p4; { long l15; p5 = cgetg(gtos(ll)+1, t_VEC); for (l15 = 1; gcmpsg(l15, ll) <= 0; ++l15) gel(p5, l15) = gen_0; } listf = p5; i = gen_1; l6 = glength(list); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long j; for (j = 1; j <= l6; ++j) { if (gequal1(gel(gel(list, j), 3))) { gel(listf, gtos(i)) = gcopy(gel(list, j)); i = gaddgs(i, 1); } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &listf, &i); } } if (!gequal0(redflag)) { l7 = glength(listf); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; for (i = 1; i <= l7; ++i) { rr = redquartic(gel(gel(listf, i), 1), prec); gel(gel(listf, i), 1) = gcopy(gel(rr, 1)); gel(gel(listf, i), 2) = gmul(gel(gel(listf, i), 2), gel(rr, 2)); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &rr, &listf); } } } if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" Output of listratpoint = %Ps\n", listf); listf = gerepilecopy(ltop, listf); return listf; } GEN redquartic(GEN pol, long prec) /* vec */ { pari_sp ltop = avma; GEN prec_s = gen_0, prec0 = gen_0, d = gen_0, disc2 = gen_0, test = gen_0, r = gen_0, normderiv = gen_0, disc2v = gen_0, q = gen_0, M = gen_0, p1 = gen_0, x = pol_x(fetch_user_var("x")); GEN p2 = gen_0, p3 = gen_0; /* vec */ /* MODI change prec into prec_s */ if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" starting redquartic\n"); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" reduction of the quartic %Ps\n", pol); /* choice of the real precision used in the computation */ prec_s = prec0 = stoi(getrealprecision()); d = stoi(degree(pol)); disc2 = gsqr(poldisc0(pol, -1)); test = gen_0; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN p4 = gen_0; /* vec */ GEN p5 = gen_0; while (gequal0(test)) { if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" precision = %Ps\n", prec_s); r = roots0(pol, 0, prec); { long i; p4 = cgetg(gtos(d)+1, t_VEC); for (i = 1; gcmpsg(i, d) <= 0; ++i) gel(p4, i) = gnorm(gsubst(deriv(pol,-1), gvar(gpolvar(pol)), gel(r, i))); } normderiv = p4; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN i = gen_0; p5 = gen_1; for (i = gen_1; gcmp(i, d) <= 0; i = gaddgs(i, 1)) { p5 = gmul(p5, gel(normderiv, gtos(i))); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &p5, &i); } } disc2v = gmul(p5, gpow(pollead(pol, -1), gsubgs(gmulsg(2, d), 4), prec)); test = stoi(gcmp(gabs(gsub(disc2v, disc2), prec), gpow(stoi(10), gdiventgs(gneg(prec_s), 2), prec)) < 0); if (gequal0(test)) setrealprecision(gtos(prec_s = gmulgs(prec_s, 2)), &prec); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 7, &r, &p4, &normderiv, &p5, &disc2v, &test, &prec_s); } } { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN i = gen_0; p1 = gen_0; for (i = gen_1; gcmp(i, d) <= 0; i = gaddgs(i, 1)) { p1 = gadd(p1, gdiv(gnorm(gsub(x, gel(r, gtos(i)))), gpow(gel(normderiv, gtos(i)), ginv(gsubgs(d, 2)), prec))); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &p1, &i); } } /* former choice of the quadratic form */ /* q = Vec(sum( i = 1, d, norm(x-r[i]))); */ /* Now, uses the quadratic form normalized as in Cremona-Stoll */ q = gtovec(p1); p2 = cgetg(3, t_MAT); gel(p2, 1) = cgetg(3, t_COL); gel(p2, 2) = cgetg(3, t_COL); gcoeff(p2, 1, 1) = gcopy(gel(q, 1)); gcoeff(p2, 1, 2) = gdivgs(gel(q, 2), 2); gcoeff(p2, 2, 1) = gdivgs(gel(q, 2), 2); gcoeff(p2, 2, 2) = gcopy(gel(q, 3)); M = QfbReduce(p2, prec); pol = gmul(gsubst(pol, gvar(gpolvar(pol)), gdiv(gtopoly(rowcopy(M, 1), -1), gtopoly(rowcopy(M, 2), -1))), gpowgs(gtopoly(rowcopy(M, 2), -1), degree(pol))); if (!gequal(prec_s, prec0)) setrealprecision(gtos(prec0), &prec); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" reduced quartic = %Ps\n", pol); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" end of redquartic\n"); p3 = cgetg(3, t_VEC); gel(p3, 1) = gcopy(pol); gel(p3, 2) = gcopy(M); p3 = gerepilecopy(ltop, p3); return p3; } GEN polrealrootsisolate(GEN pol) { pari_sp ltop = avma; GEN st = gen_0, a = gen_0, res = gen_0, ind = gen_0, b = gen_0, c = gen_0, stab = gen_0, stac = gen_0; GEN p1 = gen_0, p2 = gen_0, p3 = gen_0, p4 = gen_0; /* vec */ if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" starting polrealrootsisolate with pol = %Ps\n", pol); st = stoi(sturmpart(pol, NULL, NULL)); if (gequal0(st)) { p1 = cgetg(1, t_VEC); p1 = gerepilecopy(ltop, p1); return p1; } a = gen_1; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); while (gcmpsg(sturmpart(pol, gneg(a), a), st) < 0) { a = gshift(a, 1); if (low_stack(st_lim, stack_lim(btop, 1))) a = gerepilecopy(btop, a); } } p2 = cgetg(2, t_VEC); p3 = cgetg(4, t_VEC); gel(p3, 1) = gneg(a); gel(p3, 2) = gcopy(a); gel(p3, 3) = gcopy(st); gel(p2, 1) = p3; res = p2; ind = gen_1; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN p5 = gen_0, p6 = gen_0, p7 = gen_0; /* vec */ while (gcmpsg(glength(res), st) < 0) { { pari_sp btop = avma, st_lim = stack_lim(btop, 1); while (gequal1(gel(gel(res, gtos(ind)), 3))) { ind = gaddgs(ind, 1); if (low_stack(st_lim, stack_lim(btop, 1))) ind = gerepilecopy(btop, ind); } } a = gcopy(gel(gel(res, gtos(ind)), 1)); b = gcopy(gel(gel(res, gtos(ind)), 2)); stab = gcopy(gel(gel(res, gtos(ind)), 3)); c = gdivgs(gadd(a, b), 2); stac = stoi(sturmpart(pol, a, c)); if (gequal0(stac)) { gel(gel(res, gtos(ind)), 1) = gcopy(c); continue; } if (gequal(stac, stab)) { gel(gel(res, gtos(ind)), 2) = gcopy(c); continue; } p5 = cgetg(4, t_VEC); gel(p5, 1) = gcopy(a); gel(p5, 2) = gcopy(c); gel(p5, 3) = gcopy(stac); gel(res, gtos(ind)) = p5; p6 = cgetg(2, t_VEC); p7 = cgetg(4, t_VEC); gel(p7, 1) = gcopy(c); gel(p7, 2) = gcopy(b); gel(p7, 3) = gsub(stab, stac); gel(p6, 1) = p7; res = concat(res, p6); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 10, &ind, &a, &b, &stab, &c, &stac, &res, &p5, &p7, &p6); } } { long i; GEN p8 = gen_0; /* vec */ p4 = cgetg(gtos(st)+1, t_VEC); for (i = 1; gcmpsg(i, st) <= 0; ++i) { p8 = cgetg(3, t_VEC); gel(p8, 1) = gcopy(gel(gel(res, i), 1)); gel(p8, 2) = gcopy(gel(gel(res, i), 2)); gel(p4, i) = p8; } } res = p4; res = vecsort0(res, gen_1, 0); if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" end of polrealrootsisolate with res = %Ps\n", res); res = gerepilecopy(ltop, res); return res; } GEN polrealrootsimprove(GEN pol, GEN v) { pari_sp ltop = avma; GEN c = gen_0, v2 = gen_0, vc = gen_0; GEN p1 = gen_0; /* vec */ v = gcopy(v); c = gdivgs(gadd(gel(v, 1), gel(v, 2)), 2); v2 = gsubst(pol, gvar(gpolvar(pol)), gel(v, 2)); if (gequal0(v2)) { p1 = cgetg(3, t_VEC); gel(p1, 1) = gcopy(c); gel(p1, 2) = gcopy(gel(v, 2)); p1 = gerepilecopy(ltop, p1); return p1; } vc = gsubst(pol, gvar(gpolvar(pol)), c); if (gcmpgs(gmul(v2, vc), 0) >= 0) gel(v, 2) = gcopy(c); else gel(v, 1) = gcopy(c); v = gerepilecopy(ltop, v); return v; } GEN polrootsmodpn(GEN pol, GEN p, long prec) { pari_sp ltop = avma; GEN vd = gen_0, rac = gen_0, i = gen_0, pol2 = gen_0, r = gen_0, newrac = gen_0; GEN p1 = gen_0, p2 = gen_0; /* vec */ GEN x = pol_x(fetch_user_var("x")); if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" starting polrootsmodpn %Ps:%Ps\n", p, pol); vd = stoi(ggval(poldisc0(pol, -1), p)); p1 = cgetg(2, t_VEC); p2 = cgetg(3, t_VEC); gel(p2, 1) = gen_0; gel(p2, 2) = gen_0; gel(p1, 1) = p2; rac = p1; i = gen_1; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long l3; GEN p4 = gen_0; /* vec */ long l5; GEN p6 = gen_0; /* vec */ while (gcmpgs(i, glength(rac)) <= 0) { /* if( rac[i][2] > vd, i++; next); */ if (gcmp(gel(gel(rac, gtos(i)), 2), vd) >= 0) { i = gaddgs(i, 1); continue; } pol2 = gsubst(pol, gvar(x), gadd(gel(gel(rac, gtos(i)), 1), gmul(x, gpow(p, gel(gel(rac, gtos(i)), 2), prec)))); pol2 = gdiv(pol2, content(pol2)); r = lift(polratroots(gmul(pol2, gmodulsg(1, p)))); if (glength(r) == 0) { i = gaddgs(i, 1); continue; } l3 = glength(r); { long j; GEN p7 = gen_0; /* vec */ p4 = cgetg(l3+1, t_VEC); for (j = 1; j <= l3; ++j) { p7 = cgetg(3, t_VEC); gel(p7, 1) = gadd(gel(gel(rac, gtos(i)), 1), gmul(gpow(p, gel(gel(rac, gtos(i)), 2), prec), gel(r, j))); gel(p7, 2) = gaddgs(gel(gel(rac, gtos(i)), 2), 1); gel(p4, j) = p7; } } newrac = p4; l5 = glength(r) - 1; { long j; p6 = cgetg(l5+1, t_VEC); for (j = 1; j <= l5; ++j) gel(p6, j) = gcopy(gel(newrac, j + 1)); } rac = concat(rac, p6); gel(rac, gtos(i)) = gcopy(gel(newrac, 1)); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 7, &i, &pol2, &r, &p4, &newrac, &p6, &rac); } } if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" end of polrootsmodpn %Ps\n", rac); rac = gerepilecopy(ltop, rac); return rac; } GEN ppinit(GEN nf, GEN p) { pari_sp ltop = avma; GEN pdec = gen_0, pp = gen_0; long l1; GEN p2 = gen_0; /* vec */ pdec = idealprimedec(nf, p); l1 = glength(pdec); { long i; GEN p3 = gen_0; /* vec */ GEN p4 = gen_0; p2 = cgetg(l1+1, t_VEC); for (i = 1; i <= l1; ++i) { p3 = cgetg(5, t_VEC); gel(p3, 1) = gcopy(gel(pdec, i)); gel(p3, 2) = basistoalg(nf, gel(gel(pdec, i), 2)); if (gequalgs(p, 2)) p4 = idealstar0(nf, idealpow0(nf, gel(pdec, i), stoi(1 + (2*pr_get_e(gel(pdec, i)))), 0), 1); gel(p3, 3) = p4; gel(p3, 4) = nfmodprinit(nf, gel(pdec, i)); gel(p2, i) = p3; } } pp = p2; pp = gerepilecopy(ltop, pp); return pp; } long nfpsquareoddQ(GEN nf, GEN a, GEN pr, long prec) { pari_sp ltop = avma; GEN p = gen_0, v = gen_0, ap = gen_0, den = gen_0, norme = gen_0; if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" starting nfpsquareoddQ(%Ps,%Ps\n", a, pr); if (gequal0(a)) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" end of nfpsquareoddQ\n"); avma = ltop; return 1; } p = gcopy(gel(pr, 3)); v = stoi(idealval(nf, lift(a), p)); if (!gequal0(gmodgs(v, 2))) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" end of nfpsquareoddQ\n"); avma = ltop; return 0; } ap = algtobasis(nf, gdiv(a, gpow(basistoalg(nf, gel(p, 2)), v, prec))); den = stoi(ggval(denom(content(ap)), member_p(p))); if (!gequal0(den)) { den = gadd(den, gmodgs(den, 2)); ap = gmul(gpow(member_p(p), den, prec), nfmul(nf, ap, nfpow(nf, gel(p, 2), gmulgs(gneg(den), pr_get_e(p))))); } norme = gdivgs(gsubgs(gpowgs(member_p(p), pr_get_f(p)), 1), 2); ap = nfpowmodpr(nf, ap, norme, pr); gel(ap, 1) = gsubgs(gel(ap, 1), 1); if (gequal0(ap)) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" end of nfpsquareoddQ\n"); avma = ltop; return 1; } if (idealval(nf, ap, p) > 0) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" end of nfpsquareoddQ\n"); avma = ltop; return 1; } if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" end of nfpsquareoddQ\n"); avma = ltop; return 0; } long psquare(GEN a, GEN p, long prec) { pari_sp ltop = avma; GEN v = gen_0, ap = gen_0; GEN p1 = gen_0; /* vec */ long l2; /* bool */ if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) { p1 = cgetg(3, t_VEC); gel(p1, 1) = gcopy(a); gel(p1, 2) = gcopy(p); pari_printf(" starting psquare %Ps\n", p1); } if (gequal0(a)) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" end of psquare 1\n"); avma = ltop; return 1; } v = stoi(ggval(a, p)); if (!gequal0(gmodgs(v, 2))) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" end of psquare 0\n"); avma = ltop; return 0; } if (gequalgs(p, 2)) ap = gsubgs(gmodgs(gshift(a, -gtos(v)), 8), 1); else ap = gsubgs(gkronecker(gdiv(a, gpow(p, v, prec)), p), 1); if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" end of psquare %ld\n", gequal0(ap)); l2 = gequal0(ap); avma = ltop; return l2; } long lemma6(GEN pol, GEN p, GEN nu, GEN xx, long prec) { pari_sp ltop = avma; GEN gx = gen_0, gpx = gen_0, lambda = gen_0, mu = gen_0; gx = gsubst(pol, gvar(gpolvar(pol)), xx); if (psquare(gx, p, prec)) { avma = ltop; return 1; } gpx = gsubst(deriv(pol,-1), gvar(gpolvar(pol)), xx); lambda = stoi(ggval(gx, p)); mu = stoi(ggval(gpx, p)); if (gcmp(lambda, gmulsg(2, mu)) > 0) { avma = ltop; return 1; } /* if( (lambda >= mu+nu) && (nu > mu), return(1)); */ if ((gcmp(lambda, gmulsg(2, nu)) >= 0) && (gcmp(mu, nu) >= 0)) { avma = ltop; return 0; } avma = ltop; return -1; } long lemma7(GEN pol, GEN nu, GEN xx, long prec) { pari_sp ltop = avma; GEN gx = gen_0, gpx = gen_0, lambda = gen_0, mu = gen_0, q = gen_0; gx = gsubst(pol, gvar(gpolvar(pol)), xx); if (psquare(gx, gen_2, prec)) { avma = ltop; return 1; } gpx = gsubst(deriv(pol,-1), gvar(gpolvar(pol)), xx); lambda = stoi(ggval(gx, gen_2)); mu = stoi(ggval(gpx, gen_2)); if (gcmp(lambda, gmulsg(2, mu)) > 0) { avma = ltop; return 1; } if (gcmp(nu, mu) > 0) { if (!gequal0(gmodgs(lambda, 2))) { avma = ltop; return -1; } q = gsub(gadd(mu, nu), lambda); if (gequal1(q)) { avma = ltop; return 1; } if (gequalgs(q, 2) && gequal1(gmodgs(gshift(gx, -gtos(lambda)), 4))) { avma = ltop; return 1; } avma = ltop; return -1; } q = gsub(lambda, gmulsg(2, nu)); if (gcmpgs(q, 0) >= 0) { avma = ltop; return 0; } if (gequalgs(q, -2) && gequal1(gmodgs(gshift(gx, -gtos(lambda)), 4))) { avma = ltop; return 0; } avma = ltop; return -1; } long zpsoluble(GEN pol, GEN p, GEN nu, GEN pnu, GEN x0, GEN pnup, long prec) { pari_sp ltop = avma; GEN result = gen_0, pol2 = gen_0, fact = gen_0, x1 = gen_0; GEN p1 = gen_0; /* vec */ GEN p2 = gen_0; long l3; if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) { p1 = cgetg(5, t_VEC); gel(p1, 1) = gcopy(pol); gel(p1, 2) = gcopy(p); gel(p1, 3) = gcopy(x0); gel(p1, 4) = gcopy(nu); pari_printf(" starting zpsoluble %Ps\n", p1); } if (gequalgs(p, 2)) result = stoi(lemma7(pol, nu, x0, prec)); else result = stoi(lemma6(pol, p, nu, x0, prec)); if (gequalgs(result, 1)) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" end of zpsoluble 1 lemma\n"); avma = ltop; return 1; } if (gequalm1(result)) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" end of zpsoluble 0 lemma\n"); avma = ltop; return 0; } pnup = gmul(pnu, p); nu = gaddgs(nu, 1); if ((gcmp(p, LIMBIGPRIME) < 0) || (gequal0(LIMBIGPRIME))) { p2 = gsubgs(p, 1); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN i = gen_0; for (i = gen_0; gcmp(i, p2) <= 0; i = gaddgs(i, 1)) { if (zpsoluble(pol, p, nu, pnup, gadd(x0, gmul(pnu, i)), gen_0, prec)) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" end of zpsoluble\n"); avma = ltop; return 1; } if (low_stack(st_lim, stack_lim(btop, 1))) i = gerepilecopy(btop, i); } } } else { pol2 = gsubst(pol, gvar(gpolvar(pol)), gadd(x0, gmul(pnu, gpolvar(pol)))); pol2 = gdiv(pol2, content(pol2)); pol2 = gmul(pol2, gmodulsg(1, p)); if (!degree(pol2)) { avma = ltop; return 0; } fact = gcopy(gel(factormod0(pol2, p, 0), 1)); l3 = glength(fact); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; for (i = 1; i <= l3; ++i) { x1 = gneg(centerlift(polcoeff0(gel(fact, i), 0, -1))); if (zpsoluble(pol, p, nu, pnup, gadd(x0, gmul(pnu, x1)), gen_0, prec)) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" end of zpsoluble\n"); avma = ltop; return 1; } if (low_stack(st_lim, stack_lim(btop, 1))) x1 = gerepilecopy(btop, x1); } } { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN i = gen_0; for (i = gen_1; gcmp(i, MAXPROB) <= 0; i = gaddgs(i, 1)) { x1 = genrand(p); if (zpsoluble(pol, p, nu, pnup, gadd(x0, gmul(pnu, x1)), gen_0, prec)) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" end of zpsoluble\n"); avma = ltop; return 1; } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &i, &x1); } } } if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) if (gcmp(p, LIMBIGPRIME) >= 0) pari_printf(" ******* probabilistic test at p = %Ps*******\n", p); if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" end of zpsoluble\n"); avma = ltop; return 0; } long qpsoluble(GEN pol, GEN p, long prec) { pari_sp ltop = avma; if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) { pari_printf(" starting qpsoluble %Ps\n", p); pari_printf(" pol = %Ps\n", pol); } if (psquare(pollead(pol, -1), p, prec)) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" end of qpsoluble 1\n"); avma = ltop; return 1; } if (psquare(polcoeff0(pol, 0, -1), p, prec)) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" end of qpsoluble 1\n"); avma = ltop; return 1; } if (zpsoluble(pol, p, gen_0, gen_1, gen_0, gen_0, prec)) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" end of qpsoluble 1\n"); avma = ltop; return 1; } if (zpsoluble(polrecip(pol), p, gen_1, p, gen_0, gen_0, prec)) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" end of qpsoluble 1\n"); avma = ltop; return 1; } if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" end of qpsoluble 0\n"); avma = ltop; return 0; } long locallysoluble(GEN pol, long prec) { pari_sp ltop = avma; GEN c = gen_0, disc0 = gen_0, plist = gen_0, p = gen_0, vc = gen_0; long l1; if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" starting locallysoluble :%Ps\n", pol); /* real place */ if ((((!smodss(degree(pol), 2)) && (gsigne(pollead(pol, -1)) < 0)) && (gsigne(polcoeff0(pol, 0, -1)) < 0)) && (sturmpart(pol, NULL, NULL) == 0)) { if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" not ELS at infinity\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" end of locallysoluble\n"); avma = ltop; return 0; } /* */ /* finite places */ /* */ pol = gmul(/* */ /* finite places */ /* */ pol, gsqr(denom(content(pol)))); c = content(pol); disc0 = poldisc0(pol, -1); plist = factor(gabs(gmulsg(2, disc0), prec)); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" list of bad primes = %Ps\n", plist); l1 = glength(gel(plist, 1)); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; for (i = 1; i <= l1; ++i) { p = gcopy(gcoeff(plist, i, 1)); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" p = %Ps\n", p); vc = stoi(ggval(c, p)); if (gcmpgs(vc, 2) >= 0) { pol = gdiv(pol, gpow(p, gmulsg(2, gdiventgs(vc, 2)), prec)); gcoeff(plist, i, 2) = gsub(gcoeff(plist, i, 2), gmulgs(gmulsg(2, gdiventgs(vc, 2)), (2*degree(pol)) - 2)); } if (((degree(pol) == 4) && !gequalgs(p, 2)) && (gcmpgs(gcoeff(plist, i, 2), 2) < 0)) continue; if (!qpsoluble(pol, p, prec)) { if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" not ELS at %Ps\n", p); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" end of locallysoluble\n"); avma = ltop; return 0; } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 4, &p, &vc, &pol, &plist); } } if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" quartic ELS : Y^2 = %Ps\n", pol); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" end of locallysoluble\n"); avma = ltop; return 1; } GEN LS2localimage(GEN nf, GEN gen, GEN pp, long prec) { pari_sp ltop = avma; GEN p = gen_0, LS2image = gen_0, ph = gen_0, ival = gen_0, delta = gen_0; long l1; GEN p2 = gen_0; long l3; GEN p4 = gen_0; GEN p5 = gen_0; /* vec */ long l6; if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" starting LS2localimage\n"); p = icopy(member_p(gel(gel(pp, 1), 1))); l1 = glength(gen); if (gequalgs(p, 2)) { l3 = glength(pp); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; p4 = gen_0; for (i = 1; i <= l3; ++i) { p4 = gaddgs(p4, (1 + lg(member_cyc(gel(gel(pp, i), 3))))-1); if (low_stack(st_lim, stack_lim(btop, 1))) p4 = gerepilecopy(btop, p4); } } p2 = p4; } else p2 = stoi(2*glength(pp)); { long l7, l8; p5 = cgetg(l1+1, t_MAT); for (l8 = 1; l8 <= l1; ++l8) { gel(p5, l8) = cgetg(gtos(p2)+1, t_COL); for (l7 = 1; gcmpsg(l7, p2) <= 0; ++l7) gcoeff(p5, l7, l8) = gen_0; } } LS2image = p5; l6 = glength(gen); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long j, l9; for (j = 1; j <= l6; ++j) { ph = cgetg(1, t_VEC); l9 = glength(pp); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; GEN p10 = gen_0, p11 = gen_0; /* vec */ for (i = 1; i <= l9; ++i) { ival = stoi(idealval(nf, gel(gen, j), gel(gel(pp, i), 1))); p10 = cgetg(2, t_VEC); gel(p10, 1) = gcopy(ival); ph = concat(ph, p10); delta = gdiv(gel(gen, j), gpow(gel(gel(pp, i), 2), ival, prec)); if (gequalgs(p, 2)) ph = concat(ph, gtrans(ideallog(nf, delta, gel(gel(pp, i), 3)))); else { p11 = cgetg(2, t_VEC); gel(p11, 1) = stoi(1 - nfpsquareoddQ(nf, delta, gel(gel(pp, i), 4), prec)); ph = concat(ph, p11); } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 5, &ival, &p10, &ph, &delta, &p11); } } gel(LS2image, j) = gtrans(ph); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 4, &ph, &ival, &delta, &LS2image); } } LS2image = gmul(LS2image, gmodulss(1, 2)); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" LS2image = %Ps\n", lift(LS2image)); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" end of LS2localimage\n"); LS2image = gerepilecopy(ltop, LS2image); return LS2image; } GEN ellhalf(GEN ell, GEN P, long prec) { pari_sp ltop = avma; GEN pol2 = gen_0, ratroots = gen_0, half = gen_0, x2 = gen_0, y2 = gen_0, P2 = gen_0; GEN p1 = gen_0, p2 = gen_0; /* vec */ long l3; GEN p4 = gen_0, p5 = gen_0, p6 = gen_0, p7 = gen_0; /* vec */ long l8; if (glength(ell) < 13) ell = smallellinit(ell); p1 = cgetg(5, t_VEC); gel(p1, 1) = stoi(4); gel(p1, 2) = gcopy(ell_get_b2(ell)); gel(p1, 3) = gmulsg(2, ell_get_b4(ell)); gel(p1, 4) = gcopy(ell_get_b6(ell)); pol2 = gtopoly(p1, -1); /* 2-division polynomial */ p2 = cgetg(2, t_VEC); gel(p2, 1) = gen_0; if (gequal(P, p2)) { ratroots = polratroots(pol2); l3 = glength(ratroots); { long i; GEN p9 = gen_0; /* vec */ p4 = cgetg(l3+1, t_VEC); for (i = 1; i <= l3; ++i) { p9 = cgetg(3, t_VEC); gel(p9, 1) = gcopy(gel(ratroots, i)); gel(p9, 2) = gdivgs(gneg(gadd(gmul(ell_get_a1(ell), gel(ratroots, i)), ell_get_a3(ell))), 2); gel(p4, i) = p9; } } half = p4; p5 = cgetg(2, t_VEC); p6 = cgetg(2, t_VEC); gel(p6, 1) = gen_0; gel(p5, 1) = p6; half = concat(p5, half); half = gerepilecopy(ltop, half); return half; } p7 = cgetg(6, t_VEC); gel(p7, 1) = gen_1; gel(p7, 2) = gen_0; gel(p7, 3) = gneg(ell_get_b4(ell)); gel(p7, 4) = gmulsg(-2, ell_get_b6(ell)); gel(p7, 5) = gneg(ell_get_b8(ell)); x2 = gtopoly(p7, -1); /* x(2P) = x2/pol2 */ half = cgetg(1, t_VEC); ratroots = polratroots(gsub(x2, gmul(gel(P, 1), pol2))); if (glength(ratroots) == 0) { half = gerepilecopy(ltop, half); return half; } l8 = glength(ratroots); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i, l10; for (i = 1; i <= l8; ++i) { y2 = ellordinate(ell, gel(ratroots, i), prec); l10 = glength(y2); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long j; GEN p11 = gen_0, p12 = gen_0; /* vec */ for (j = 1; j <= l10; ++j) { p11 = cgetg(3, t_VEC); gel(p11, 1) = gcopy(gel(ratroots, i)); gel(p11, 2) = gcopy(gel(y2, j)); P2 = p11; if (gequal(powell(ell, P2, gen_2), P)) { p12 = cgetg(2, t_VEC); gel(p12, 1) = gcopy(P2); half = concat(half, p12); } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 4, &p11, &P2, &p12, &half); } } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 3, &y2, &P2, &half); } } half = gerepilecopy(ltop, half); return half; } GEN elltors2(GEN ell, long prec) { pari_sp ltop = avma; GEN tors2 = gen_0; GEN p1 = gen_0, p2 = gen_0, p3 = gen_0, p4 = gen_0, p5 = gen_0, p6 = gen_0, p7 = gen_0, p8 = gen_0; /* vec */ if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" computing the 2-torsion\n"); p1 = cgetg(2, t_VEC); gel(p1, 1) = gen_0; tors2 = ellhalf(ell, p1, prec); if (glength(tors2) == 1) { p2 = cgetg(4, t_VEC); gel(p2, 1) = gen_1; gel(p2, 2) = cgetg(1, t_VEC); gel(p2, 3) = cgetg(1, t_VEC); tors2 = p2; } else { if (glength(tors2) == 2) { p3 = cgetg(4, t_VEC); gel(p3, 1) = gen_2; p4 = cgetg(2, t_VEC); gel(p4, 1) = gen_2; gel(p3, 2) = p4; p5 = cgetg(2, t_VEC); gel(p5, 1) = gcopy(gel(tors2, 2)); gel(p3, 3) = p5; tors2 = p3; } else { p6 = cgetg(4, t_VEC); gel(p6, 1) = stoi(4); p7 = cgetg(3, t_VEC); gel(p7, 1) = gen_2; gel(p7, 2) = gen_2; gel(p6, 2) = p7; p8 = cgetg(3, t_VEC); gel(p8, 1) = gcopy(gel(tors2, 2)); gel(p8, 2) = gcopy(gel(tors2, 3)); gel(p6, 3) = p8; tors2 = p6; } } if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" E[2] = %Ps\n", tors2); tors2 = gerepilecopy(ltop, tors2); return tors2; } GEN elltorseven(GEN ell, long prec) { pari_sp ltop = avma; GEN torseven = gen_0, P2 = gen_0; if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" computing the 2^n-torsion\n"); if (glength(ell) < 13) ell = smallellinit(ell); torseven = elltors2(ell, prec); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); while (!gequalgs(gel(torseven, 1), 1)) { P2 = ellhalf(ell, gel(gel(torseven, 3), 1), prec); if (glength(P2) > 0) { gel(torseven, 1) = gmulgs(gel(torseven, 1), 2); gel(gel(torseven, 2), 1) = gmulgs(gel(gel(torseven, 2), 1), 2); gel(gel(torseven, 3), 1) = gcopy(gel(P2, 1)); continue; } if (glength(gel(torseven, 3)) == 1) break; P2 = ellhalf(ell, gel(gel(torseven, 3), 2), prec); if (glength(P2) > 0) { gel(torseven, 1) = gmulgs(gel(torseven, 1), 2); gel(gel(torseven, 2), 2) = gmulgs(gel(gel(torseven, 2), 2), 2); gel(gel(torseven, 3), 2) = gcopy(gel(P2, 1)); continue; } P2 = ellhalf(ell, addell(ell, gel(gel(torseven, 3), 1), gel(gel(torseven, 3), 2)), prec); if (glength(P2) > 0) { gel(torseven, 1) = gmulgs(gel(torseven, 1), 2); gel(gel(torseven, 2), 1) = gmulgs(gel(gel(torseven, 2), 1), 2); gel(gel(torseven, 3), 1) = gcopy(gel(P2, 1)); continue; } break; if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &P2, &torseven); } } if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" E[2^n] = %Ps\n", torseven); torseven = gerepilecopy(ltop, torseven); return torseven; } GEN ellsort(GEN listpts, long prec) /* vec */ { pari_sp ltop = avma; GEN n = gen_0, v = gen_0, aux = gen_0, ord = gen_0, p1 = gen_0; GEN p2 = gen_0, p3 = gen_0; /* vec */ p1 = n = stoi(glength(listpts)); { long l4; p2 = cgetg(gtos(p1)+1, t_VEC); for (l4 = 1; gcmpsg(l4, p1) <= 0; ++l4) gel(p2, l4) = gen_0; } v = p2; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN i = gen_0; GEN p5 = gen_0, p6 = gen_0, p7 = gen_0; /* vec */ for (i = gen_1; gcmp(i, n) <= 0; i = gaddgs(i, 1)) { p5 = cgetg(2, t_VEC); gel(p5, 1) = gen_0; if (gequal(gel(listpts, gtos(i)), p5)) { p6 = cgetg(4, t_VEC); gel(p6, 1) = gen_0; gel(p6, 2) = gen_0; gel(p6, 3) = gen_0; gel(v, gtos(i)) = p6; continue; } aux = gdiv(denom(gel(gel(listpts, gtos(i)), 2)), denom(gel(gel(listpts, gtos(i)), 1))); p7 = cgetg(4, t_VEC); gel(p7, 1) = gmul(gel(gel(listpts, gtos(i)), 1), gsqr(aux)); gel(p7, 2) = gmul(gel(gel(listpts, gtos(i)), 2), gpowgs(aux, 3)); gel(p7, 3) = gcopy(aux); gel(v, gtos(i)) = vecsort0(gabs(p7, prec), NULL, 4); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 6, &i, &p5, &p6, &v, &aux, &p7); } } ord = gtovec(vecsort0(v, NULL, 3)); /* MODI Needed for gp2c */ { long i; p3 = cgetg(gtos(n)+1, t_VEC); for (i = 1; gcmpsg(i, n) <= 0; ++i) gel(p3, i) = gcopy(gel(listpts, gtos(gel(ord, i)))); } p3 = gerepilecopy(ltop, p3); return p3; } GEN ellremovetorsion(GEN ell, GEN listgen) { pari_sp ltop = avma; GEN d = gen_0, extra = gen_0; if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" removing torsion from %Ps\n", listgen); d = stoi(glength(listgen)); extra = gen_0; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN i = gen_0; GEN p1 = gen_0, p2 = gen_0; /* vec */ for (i = gen_1; gcmp(i, d) <= 0; i = gaddgs(i, 1)) { /* points of order 1 or 2 */ p1 = cgetg(2, t_VEC); gel(p1, 1) = gen_0; if (gequal(gel(listgen, gtos(i)), p1) || gequal(gel(listgen, gtos(i)), powell(ell, gel(listgen, gtos(i)), gen_m1))) { extra = gadd(extra, shifti(gen_1, gtos(gsubgs(i, 1)))); continue; } /* detection of infinite order points by looking at */ /* 8*9*5*7*P modulo the prime 1048583 */ p2 = cgetg(2, t_VEC); gel(p2, 1) = gen_0; if ((!gequalgs(gmodgs(member_disc(ell), 1048583), 0) && !gequalgs(gmodgs(denom(gel(listgen, gtos(i))), 1048583), 0)) && !gequal(powell(ell, gmul(gel(listgen, gtos(i)), gmodulss(1, 1048583)), stoi(2520)), p2)) continue; /* detection of torsion points by ellorder() */ if (!gequal0(ellorder(ell, gel(listgen, gtos(i)), NULL))) extra = gadd(extra, shifti(gen_1, gtos(gsubgs(i, 1)))); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 4, &i, &p1, &extra, &p2); } } if (!gequal0(extra)) listgen = extract0(listgen, gsub(subis(shifti(gen_1, glength(listgen)), 1), extra), NULL); if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" without torsion = %Ps\n", listgen); listgen = gerepilecopy(ltop, listgen); return listgen; } GEN ellredgen(GEN ell0, GEN listgen, GEN K, long prec) { pari_sp ltop = avma; GEN d = gen_0, ell = gen_0, sqrtK = gen_0, urst = gen_0, extra = gen_0, M = gen_0, U = gen_0, listgen2 = gen_0, tors2 = gen_0, vt = gen_0; GEN p1 = gen_0, p2 = gen_0, p3 = gen_0, p4 = gen_0, p5 = gen_0; /* vec */ GEN p6 = gen_0; GEN p7 = gen_0; /* vec */ long l8; GEN p9 = gen_0; /* vec */ if (!K) K = gen_1; listgen = gcopy(listgen); ell = gcopy(ell0); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" Reduction of the generators %Ps\n", listgen); if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" ell=%Ps\n", ell); d = stoi(glength(listgen)); if (gequal0(d)) { p1 = cgetg(1, t_VEC); p1 = gerepilecopy(ltop, p1); return p1; } /* removing torsion points from listgen */ listgen = ellremovetorsion(ell0, listgen); d = stoi(glength(listgen)); if (gequal0(d)) { p2 = cgetg(1, t_VEC); p2 = gerepilecopy(ltop, p2); return p2; } if (glength(ell) < 13) ell = smallellinit(ell); if (!gequalgs(K, 1)) { if (!gequalgs(ell_get_a1(ell), 0) || !gequalgs(ell_get_a3(ell), 0)) pari_err(user, " ellredgen : a1*a3 != 0"); gel(ell, 2) = gmul(gel(ell, 2), K); gel(ell, 4) = gmul(gel(ell, 4), gsqr(K)); gel(ell, 5) = gmul(gel(ell, 5), gpowgs(K, 3)); gel(ell, 6) = gmul(gel(ell, 6), K); gel(ell, 7) = gmul(gel(ell, 7), gsqr(K)); gel(ell, 8) = gmul(gel(ell, 8), gpowgs(K, 3)); gel(ell, 9) = gmul(gel(ell, 9), gpowgs(K, 4)); gel(ell, 10) = gmul(gel(ell, 10), gsqr(K)); gel(ell, 11) = gmul(gel(ell, 11), gpowgs(K, 3)); gel(ell, 12) = gmul(gel(ell, 12), gpowgs(K, 6)); sqrtK = gsqrt(K, prec); if (glength(ell) == 19) { gel(ell, 14) = gmul(gel(ell, 14), K); gel(ell, 15) = gdiv(gel(ell, 15), sqrtK); gel(ell, 16) = gdiv(gel(ell, 16), sqrtK); gel(ell, 17) = gmul(gel(ell, 17), sqrtK); gel(ell, 18) = gmul(gel(ell, 18), sqrtK); gel(ell, 19) = gdiv(gel(ell, 19), K); } { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN i = gen_0; long l10; for (i = gen_1; gcmp(i, d) <= 0; i = gaddgs(i, 1)) { l10 = glength(gel(listgen, gtos(i))); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long j; for (j = 1; j <= l10; ++j) { gel(gel(listgen, gtos(i)), j) = gmul(gel(gel(listgen, gtos(i)), j), gpowgs(K, j)); if (low_stack(st_lim, stack_lim(btop, 1))) listgen = gerepilecopy(btop, listgen); } } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &i, &listgen); } } } if (gequal1(d)) { p3 = cgetg(5, t_VEC); gel(p3, 1) = gen_1; gel(p3, 2) = gen_0; gel(p3, 3) = gen_0; gel(p3, 4) = gen_0; urst = p3; } else { if (glength(ell) < 19) ell = ellinit(ell, prec); ell = ellminimalmodel(ell, &urst); listgen = ellchangepoint(listgen, urst); if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" ell = %Ps\n", ell); if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" listgen = %Ps\n", listgen); /* Looking for relations between the points in listgen */ /* using LLL on the height matrix */ extra = gen_1; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN p11 = gen_0; /* vec */ while (!gequal0(extra)) { M = mathell(ell, listgen, prec); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" height matrix = %Ps\n", M); if (gcmp(gabs(det(M), prec), gpowgs(stoi(10), (-getrealprecision()) + 10)) > 0) break; U = lllkerim(ground(gmul(M, gpowgs(stoi(10), getrealprecision() - 10)))); U = concat(gel(U, 1), gel(U, 2)); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" change of basis proposed by LLL = %Ps\n", U); /* the columns of U that have very small coefficients */ /* are either exact relations or reductions (coeff <= 20) */ /* the other ones are irrelevant. */ extra = gen_0; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN i = gen_0; for (i = gen_1; gcmp(i, d) <= 0; i = gaddgs(i, 1)) { if (gcmpgs(vecmax(gabs(gel(U, gtos(i)), prec)), 20) > 0) extra = gadd(extra, shifti(gen_1, gtos(gsubgs(i, 1)))); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &i, &extra); } } U = extract0(U, gsub(subis(shifti(gen_1, gtos(d)), 1), extra), NULL); U = completebasis(U, NULL); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" change of basis 1 = %Ps\n", U); { long l12; p11 = cgetg(gtos(d)+1, t_VEC); for (l12 = 1; gcmpsg(l12, d) <= 0; ++l12) gel(p11, l12) = gen_0; } listgen2 = p11; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN i = gen_0; GEN p13 = gen_0; /* vec */ for (i = gen_1; gcmp(i, d) <= 0; i = gaddgs(i, 1)) { p13 = cgetg(2, t_VEC); gel(p13, 1) = gen_0; gel(listgen2, gtos(i)) = p13; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN j = gen_0; for (j = gen_1; gcmp(j, d) <= 0; j = gaddgs(j, 1)) { gel(listgen2, gtos(i)) = addell(ell, gel(listgen2, gtos(i)), powell(ell, gel(listgen, gtos(j)), gcoeff(U, gtos(j), gtos(i)))); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &j, &listgen2); } } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 3, &i, &p13, &listgen2); } } listgen = gcopy(listgen2); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 6, &M, &U, &extra, &p11, &listgen2, &listgen); } } /* Extracting the points of infinite order */ /* removing torsion points from listgen */ listgen = ellremovetorsion(ell, listgen); d = stoi(glength(listgen)); if (gequal0(d)) { p4 = cgetg(1, t_VEC); p4 = gerepilecopy(ltop, p4); return p4; } } if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" infinite order points = %Ps\n", listgen); /* Now, the points should be of infinite order and independant */ /* Reducing the points of infinite order */ if (gcmpgs(d, 1) > 0) { M = mathell(ell, listgen, prec); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" height matrix = %Ps\n", M); U = qflllgram0(M, 0); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" change of basis 2 = %Ps\n", U); { long l14; p5 = cgetg(gtos(d)+1, t_VEC); for (l14 = 1; gcmpsg(l14, d) <= 0; ++l14) gel(p5, l14) = gen_0; } listgen2 = p5; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN i = gen_0; GEN p15 = gen_0; /* vec */ for (i = gen_1; gcmp(i, d) <= 0; i = gaddgs(i, 1)) { p15 = cgetg(2, t_VEC); gel(p15, 1) = gen_0; gel(listgen2, gtos(i)) = p15; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN j = gen_0; for (j = gen_1; gcmp(j, d) <= 0; j = gaddgs(j, 1)) { gel(listgen2, gtos(i)) = addell(ell, gel(listgen2, gtos(i)), powell(ell, gel(listgen, gtos(j)), gcoeff(U, gtos(j), gtos(i)))); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &j, &listgen2); } } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 3, &i, &p15, &listgen2); } } listgen = gcopy(listgen2); } if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" infinite order points = %Ps\n", listgen); listgen = ellchangepointinverse(listgen, urst); /* Reducing modulo the 2-torsion */ tors2 = elltorseven(ell0, prec); if (gcmpgs(gel(tors2, 1), 1) > 0) { p6 = gcopy(gel(gel(tors2, 2), 1)); { long j; p7 = cgetg(gtos(p6)+1, t_VEC); for (j = 1; gcmpsg(j, p6) <= 0; ++j) gel(p7, j) = powell(ell0, gel(gel(tors2, 3), 1), stoi(j - 1)); } vt = p7; if (glength(gel(tors2, 2)) == 2) { l8 = glength(vt); { long j; p9 = cgetg(l8+1, t_VEC); for (j = 1; j <= l8; ++j) gel(p9, j) = addell(ell0, gel(vt, j), gel(gel(tors2, 3), 2)); } vt = concat(vt, p9); } { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN i = gen_0; long l16; GEN p17 = gen_0; /* vec */ for (i = gen_1; gcmp(i, d) <= 0; i = gaddgs(i, 1)) { l16 = glength(vt); { long j; p17 = cgetg(l16+1, t_VEC); for (j = 1; j <= l16; ++j) gel(p17, j) = addell(ell0, gel(listgen, gtos(i)), gel(vt, j)); } gel(listgen, gtos(i)) = gcopy(gel(ellsort(p17, prec), 1)); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 3, &i, &p17, &listgen); } } } if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" infinite order points = %Ps\n", listgen); if (!gequalgs(K, 1)) { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN i = gen_0; for (i = gen_1; gcmp(i, d) <= 0; i = gaddgs(i, 1)) { { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long j; for (j = 1; j <= 2; ++j) { gel(gel(listgen, gtos(i)), j) = gdiv(gel(gel(listgen, gtos(i)), j), gpowgs(K, j)); if (low_stack(st_lim, stack_lim(btop, 1))) listgen = gerepilecopy(btop, listgen); } } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &i, &listgen); } } /* keep only the points (x,y) with y >= 0 */ if (gequal0(ell_get_a1(ell0)) && gequal0(ell_get_a3(ell0))) { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN i = gen_0; for (i = gen_1; gcmp(i, d) <= 0; i = gaddgs(i, 1)) { if (glength(gel(listgen, gtos(i))) == 2) gel(gel(listgen, gtos(i)), 2) = gabs(gel(gel(listgen, gtos(i)), 2), prec); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &i, &listgen); } } if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" reduced generators = %Ps\n", listgen); listgen = gerepilecopy(ltop, listgen); return listgen; } GEN reducemodsquares(GEN delta, GEN d, long prec) { pari_sp ltop = avma; GEN deg = gen_0, xx = gen_0, z = gen_0, qd = gen_0, Qd = gen_0, reduc = gen_0, x = pol_x(fetch_user_var("x")); GEN p1 = gen_0, p2 = gen_0; /* vec */ GEN p3 = gen_0; deg = stoi(degree(compo(delta, 1))); /* MODI to make gp2c happy */ xx = gmodulo(x, compo(delta, 1)); { long i; p1 = cgetg(gtos(deg)+1, t_VEC); for (i = 1; gcmpsg(i, deg) <= 0; ++i) gel(p1, i) = geval(concat(strtoGENstr("a"), stoi(i))); } z = gsubst(gtopoly(p1, -1), gvar(x), xx); qd = polcoeff0(lift(gmul(delta, gsqr(z))), gtos(d), gvar(x)); { long i, j; p2 = cgetg(gtos(deg)+1, t_MAT); for (j = 1; gcmpsg(j, deg) <= 0; ++j) { gel(p2, j) = cgetg(gtos(deg)+1, t_COL); for (i = 1; gcmpsg(i, deg) <= 0; ++i) gcoeff(p2, i, j) = gdivgs(deriv(deriv(qd, gvar(geval(concat(strtoGENstr("a"), stoi(i))))), gvar(geval(concat(strtoGENstr("a"), stoi(j))))), 2); } } Qd = simplify(p2); reduc = IndefiniteLLL(Qd, NULL, NULL, prec); if (glength(reduc) == 2) reduc = gcopy(gel(gel(reduc, 2), 1)); p3 = gmul(delta, gsqr(gsubst(gtopoly(reduc, -1), gvar(x), xx))); p3 = gerepilecopy(ltop, p3); return p3; } GEN bnfpSelmer(GEN bnf, GEN S, GEN p, long prec) /* vec */ { pari_sp ltop = avma; GEN S1 = gen_0, oddclass = gen_0, multS = gen_0, Slist = gen_0, LS2gen = gen_0, newprimes = gen_0, newprimesval = gen_0, kerval = gen_0; long l1, l2, l3; GEN p4 = gen_0; /* vec */ long l5; GEN p6 = gen_0, p7 = gen_0, p8 = gen_0; /* vec */ if (!S) S = gen_1; if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" Constructing the field Selmer group : L(S,%Ps)\n", p); S1 = idealhnf0(bnf, S, NULL); oddclass = gen_0; multS = gen_1; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); while (gequal0(oddclass)) { if (!gequalgs(multS, 1)) S1 = idealmul(bnf, S1, multS); Slist = gtrans(gel(idealfactor(bnf, S1), 1)); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" constructing the S-units \n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" S1 = %Ps\n", Slist); LS2gen = bnfsunit(bnf, Slist, prec); /* If the class group is divisible by p, */ /* need to enlarge S1. */ oddclass = gmod(member_no(gel(LS2gen, 5)), p); if (gequal0(oddclass)) { if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" class group divisible by p = %Ps\n", member_no(gel(LS2gen, 5))); multS = idealmul(bnf, S, gel(member_gen(gel(LS2gen, 5)), 1)); } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 5, &S1, &Slist, &LS2gen, &oddclass, &multS); } } LS2gen = gmodulo(gel(LS2gen, 1), member_pol(bnf)); /* The valuation of the generators must be divisible by p outside S. */ newprimes = cgetg(1, t_VEC); l1 = glength(Slist); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; GEN p9 = gen_0; /* vec */ for (i = 1; i <= l1; ++i) { if (gequal1(idealadd(bnf, S, gel(Slist, i)))) { p9 = cgetg(2, t_VEC); gel(p9, 1) = gcopy(gel(Slist, i)); newprimes = concat(newprimes, p9); } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &p9, &newprimes); } } if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" newprimes = %Ps\n", newprimes); l2 = glength(LS2gen); l3 = glength(newprimes); { long i, j; p4 = cgetg(l2+1, t_MAT); for (j = 1; j <= l2; ++j) { gel(p4, j) = cgetg(l3+1, t_COL); for (i = 1; i <= l3; ++i) gcoeff(p4, i, j) = stoi(idealval(bnf, gel(LS2gen, j), gel(newprimes, i))); } } newprimesval = p4; if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" newprimesval = %Ps\n", newprimesval); kerval = lift(matker0(gmul(newprimesval, gmodulsg(1, p)), 0)); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" kerval = %Ps\n", kerval); l5 = glength(kerval); { long i, l10; GEN p11 = gen_0; p6 = cgetg(l5+1, t_VEC); for (i = 1; i <= l5; ++i) { l10 = glength(LS2gen); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long j; p11 = gen_1; for (j = 1; j <= l10; ++j) { p11 = gmul(p11, gpow(gel(LS2gen, j), gcoeff(kerval, j, i), prec)); if (low_stack(st_lim, stack_lim(btop, 1))) p11 = gerepilecopy(btop, p11); } } gel(p6, i) = p11; } } LS2gen = p6; /* Add the units */ LS2gen = concat(member_fu(bnf), LS2gen); /* Add also the torsion unit if its order is divisible by p. */ if (gequal0(gmod(gel(member_tu(bnf), 1), p))) { p7 = cgetg(2, t_VEC); gel(p7, 1) = gcopy(gel(member_tu(bnf), 2)); LS2gen = concat(p7, LS2gen); } if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" #LS2gen = %ld\n", glength(LS2gen)); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" LS2gen = %Ps\n", LS2gen); p8 = cgetg(3, t_VEC); gel(p8, 1) = gcopy(LS2gen); gel(p8, 2) = gcopy(Slist); p8 = gerepilecopy(ltop, p8); return p8; } GEN kersign(GEN gen, GEN rootapprox) { pari_sp ltop = avma; GEN signs = gen_0, elt = gen_0, elt2 = gen_0, d = gen_0, st = gen_0, kers = gen_0, compt = gen_0; long l1; GEN p2 = gen_0; /* vec */ long l3; gen = gcopy(gen); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" Computing the kernel of the sign %Ps\n", rootapprox); l1 = glength(gen); { long l4; p2 = cgetg(l1+1, t_VEC); for (l4 = 1; l4 <= l1; ++l4) gel(p2, l4) = gen_0; } /* determination of the signs */ signs = p2; l3 = glength(gen); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; GEN p5 = gen_0; for (i = 1; i <= l3; ++i) { elt = lift(gel(gen, i)); if (degree(elt) == 0) { gel(signs, i) = stoi(gsigne(simplify(elt)) < 0); continue; } d = poldisc0(elt, -1); if (degree(elt) == 2) if (gcmpgs(d, 0) <= 0) { gel(signs, i) = stoi(gsigne(pollead(elt, -1)) < 0); continue; } if (gequal0(d)) p5 = gdiv(elt, ggcd(elt, deriv(elt,-1))); else p5 = elt; elt2 = p5; st = gen_1; compt = gen_0; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); while (!gequal0(st)) { st = stoi(sturmpart(elt2, gel(rootapprox, 1), gel(rootapprox, 2))); if (!gequal0(st)) { rootapprox = polrealrootsimprove(compo(gel(gen, i), 1), rootapprox); /* MODI to make gp2c happy */ /* if the sign of elt is too difficult to determine, */ /* try the sign of 1/elt. */ if (gequal0(gmodgs(compt = gaddgs(compt, 1), 5))) { gel(gen, i) = ginv(gel(gen, i)); --i; goto label2; } } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 4, &st, &rootapprox, &compt, &gen); } } gel(signs, i) = stoi(gsigne(gsubst(elt, gvar(gpolvar(elt)), gel(rootapprox, 2))) < 0); label2:; if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 9, &p5, &elt, &signs, &d, &elt2, &st, &compt, &rootapprox, &gen); } } if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" signs = %Ps\n", signs); /* construction of the kernel */ kers = gmul(matker0(gtomat(gmul(signs, gmodulss(1, 2))), 0), gmodulss(1, 2)); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" kers = %Ps\n", lift(kers)); kers = gerepilecopy(ltop, kers); return kers; } GEN kernorm(GEN gen, GEN S, GEN p) { pari_sp ltop = avma; GEN normgen = gen_0, normmap = gen_0, kern = gen_0; GEN p1 = gen_0; /* vec */ long l2, l3; GEN p4 = gen_0; /* vec */ if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" Computing the kernel of the norm map\n"); if (gequalgs(p, 2)) { p1 = cgetg(2, t_VEC); gel(p1, 1) = gen_m1; S = concat(p1, S); } normgen = gnorm(gen); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" normgen = %Ps\n", normgen); l2 = glength(normgen); l3 = glength(S); { long i, j, l5 = 0; p4 = cgetg(l2+1, t_MAT); for (j = 1; j <= l2; ++j) { gel(p4, j) = cgetg(l3+1, t_COL); for (i = 1; i <= l3; ++i) { if ((i == 1) && gequalgs(p, 2)) l5 = gsigne(gel(normgen, j)) < 0; else l5 = ggval(gel(normgen, j), gel(S, i)); gcoeff(p4, i, j) = stoi(l5); } } } /* matrix of the norm map */ normmap = p4; if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" normmap = %Ps\n", normmap); /* construction of the kernel */ kern = gmul(matker0(gmul(normmap, gmodulsg(1, p)), 0), gmodulsg(1, p)); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" ker = %Ps\n", lift(kern)); kern = gerepilecopy(ltop, kern); return kern; } GEN elllocalimage(GEN nf, GEN pp, GEN K, long prec) { pari_sp ltop = avma; GEN X = gen_0, p = gen_0, prank = gen_0, rac = gen_0, pts = gen_0, bound = gen_0, essai = gen_0, mrank = gen_0, r = gen_0, xx = gen_0, delta = gen_0, ph = gen_0, delta2 = gen_0, prec_s = gen_0, ival = gen_0, x = pol_x(fetch_user_var("x")); GEN p1 = gen_0; /* vec */ if (!K) K = gen_1; /* MODI add prec_s, ival */ if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" starting elllocalimage\n"); X = gmodulo(x, member_pol(nf)); p = gcopy(gel(gel(gel(pp, 1), 1), 1)); prank = stoi(glength(pp) - !gequalgs(p, 2)); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" prank = %Ps\n", prank); rac = polrootsmodpn(gmul(K, member_pol(nf)), p, prec); if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" rac = %Ps\n", rac); { long l2, l3; p1 = cgetg(1, t_MAT); for (l3 = 1; l3 <= 0; ++l3) { gel(p1, l3) = cgetg(1, t_COL); for (l2 = 1; l2 <= 0; ++l2) gcoeff(p1, l2, l3) = gen_0; } } pts = p1; bound = gaddgs(p, 6); essai = gen_0; mrank = gen_0; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long l4; while (gcmp(mrank, prank) < 0) { essai = gaddgs(essai, 1); if (gequal0(gmodgs(essai, 16))) { pts = matimage0(pts, 0); bound = gmul(bound, p); } r = gaddgs(genrand(stoi(glength(rac))), 1); prec_s = gsubgs(genrand(gaddgs(gel(gel(rac, gtos(r)), 2), 3)), 2); xx = gadd(gel(gel(rac, gtos(r)), 1), gmul(gpow(p, prec_s, prec), genrand(bound))); if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" xx = %Ps\n", xx); delta = gmul(K, gsub(xx, X)); /* rem : K*pol(xx) = norm(delta) ( = y^2 for a point on the elliptic curve) */ if (!psquare(gmul(K, gsubst(member_pol(nf), gvar(x), xx)), p, prec)) continue; ph = cgetg(1, t_VEC); l4 = glength(pp); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; GEN p5 = gen_0, p6 = gen_0; /* vec */ for (i = 1; i <= l4; ++i) { p5 = cgetg(2, t_VEC); gel(p5, 1) = gcopy(ival = stoi(idealval(nf, delta, gel(gel(pp, i), 1)))); ph = concat(ph, p5); delta2 = gdiv(delta, gpow(gel(gel(pp, i), 2), ival, prec)); if (gequalgs(p, 2)) ph = concat(ph, gtrans(ideallog(nf, delta2, gel(gel(pp, i), 3)))); else { p6 = cgetg(2, t_VEC); gel(p6, 1) = stoi(1 - nfpsquareoddQ(nf, delta2, gel(gel(pp, i), 4), prec)); ph = concat(ph, p6); } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 5, &ival, &p5, &ph, &delta2, &p6); } } if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" ph = %Ps\n", ph); pts = concat(pts, gmul(gtrans(ph), gmodulss(1, 2))); mrank = stoi(rank(gmul(pts, gmodulss(1, 2)))); if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" pts = %Ps\n", lift(pts)); if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" matrank = %Ps\n", mrank); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 11, &essai, &pts, &bound, &r, &prec_s, &xx, &delta, &ph, &ival, &delta2, &mrank); } } pts = matimage0(pts, 0); if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" essai = %Ps\n", essai); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" end of elllocalimage\n"); pts = gerepilecopy(ltop, pts); return pts; } GEN ell2descent_gen(GEN ell, GEN bnf, GEN K, GEN help, GEN redflag, long prec) /* vec */ { pari_sp ltop = avma; GEN A = gen_0, B = gen_0, C = gen_0, polrel = gen_0, polprime = gen_0, ttheta = gen_0, badprimes = gen_0, S = gen_0, LS2 = gen_0, selmer = gen_0, rootapprox = gen_0, p = gen_0, pp = gen_0, locimage = gen_0, LS2image = gen_0, listpointstriv = gen_0, listpoints = gen_0, iwhile = gen_0, expo = gen_0, zc = gen_0, liftzc = gen_0, den = gen_0, point = gen_0, idealfactorzc = gen_0, idealzc = gen_0, baseidealzc = gen_0, q2 = gen_0, sol = gen_0, param = gen_0, q1 = gen_0, pol = gen_0, redq = gen_0, q0 = gen_0, pointxx = gen_0, point2 = gen_0, rang = gen_0; GEN p1 = gen_0; /* vec */ GEN x = pol_x(fetch_user_var("x")); long l2; GEN p3 = gen_0; /* vec */ long l4, l5; GEN y = pol_x(fetch_user_var("y")); long l6; GEN p7 = gen_0; /* vec */ if (!K) K = gen_1; if (!help) help = cgetg(1, t_VEC); if (!redflag) redflag = gen_0; if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" starting ell2descent_gen\n"); if (glength(ell) < 13) ell = smallellinit(ell); if (!gequalgs(ell_get_a1(ell), 0) || !gequalgs(ell_get_a3(ell), 0)) pari_err(user, " ell2descent_gen : the curve is not of the form [0,a,0,b,c]"); if (((gcmpgs(denom(ell_get_a2(ell)), 1) > 0) || (gcmpgs(denom(ell_get_a4(ell)), 1) > 0)) || (gcmpgs(denom(ell_get_a6(ell)), 1) > 0)) pari_err(user, " ell2descent_gen : non integral coefficients"); A = gcopy(ell_get_a2(ell)); if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" A = %Ps\n", A); B = gcopy(ell_get_a4(ell)); if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" B = %Ps\n", B); C = gcopy(ell_get_a6(ell)); if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" C = %Ps\n", C); /* \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ */ /* Construction of L(S,2) \\ */ /* \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ */ if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) { pari_printf("\n"); pari_printf(" Computing L(S,2)\n"); } p1 = cgetg(5, t_VEC); gel(p1, 1) = gen_1; gel(p1, 2) = gcopy(A); gel(p1, 3) = gcopy(B); gel(p1, 4) = gcopy(C); polrel = gtopoly(p1, -1); polprime = deriv(polrel,-1); ttheta = gmodulo(x, polrel); if (gequal0(bnf)) { if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" bnfinit(%Ps)\n", polrel); bnf = Buchall(polrel, nf_FORCE, prec); } badprimes = gabs(gmul(K, idealadd(bnf, polprime, member_index(bnf))), prec); if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" badprimes = %Ps\n", gcoeff(badprimes, 1, 1)); S = bnfpSelmer(bnf, badprimes, gen_2, prec); LS2 = gcopy(gel(S, 1)); S = gcopy(gel(S, 2)); if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" L(S,2) = %Ps\n", LS2); /* \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ */ /* Construction of the Selmer group \\ */ /* \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ */ if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) { pari_printf("\n"); pari_printf(" Computing the Selmer group\n"); } l2 = glength(S); { long i; p3 = cgetg(l2+1, t_VEC); for (i = 1; i <= l2; ++i) gel(p3, i) = icopy(member_p(gel(S, i))); } /* elements with square norm */ selmer = kernorm(LS2, p3, gen_2); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" selmer = %Ps\n", lift(selmer)); /* the first real embedding must be > 0 */ /* since the norm is a square, this is automatic */ /* if there is a single real embedding. */ if (cmpis(member_r1(bnf), 3) == 0) { rootapprox = gcopy(gel(polrealrootsisolate(polrel), 1)); selmer = gmul(intersect(selmer, kersign(LS2, rootapprox)), gmodulss(1, 2)); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" selmer = %Ps\n", lift(selmer)); } /* p-adic points */ if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" p-adic points\n"); badprimes = gcopy(gel(factorint(gmulgs(gcoeff(badprimes, 1, 1), 2), 0), 1)); if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" badprimes = %Ps\n", badprimes); l4 = glength(badprimes); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; for (i = 1; i <= l4; ++i) { p = gcopy(gel(badprimes, i)); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" p = %Ps\n", p); pp = ppinit(member_nf(bnf), p); locimage = elllocalimage(member_nf(bnf), pp, K, prec); LS2image = LS2localimage(member_nf(bnf), LS2, pp, prec); locimage = intersect(LS2image, locimage); selmer = intersect(selmer, concat(matker0(LS2image, 0), gmul(inverseimage(LS2image, locimage), gmodulss(1, 2)))); selmer = matimage0(gmul(selmer, gmodulss(1, 2)), 0); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" selmer = %Ps\n", selmer); if (!glength(selmer)) break; if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 5, &p, &pp, &locimage, &LS2image, &selmer); } } if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" selmer = %Ps\n", lift(selmer)); if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" Selmer rank = %ld\n", glength(selmer)); /* \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ */ /* Search for trivial points \\ */ /* \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ */ if (glength(selmer)) { if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) { pari_printf("\n"); pari_printf(" Search for trivial points on the curve\n"); } listpointstriv = ratpoint(gmul(gpowgs(K, 3), gsubst(polrel, gvar(x), gdiv(x, K))), LIMTRIV, gen_0, prec); l5 = glength(listpointstriv); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; GEN p8 = gen_0; /* vec */ for (i = 1; i <= l5; ++i) { if (glength(gel(listpointstriv, i)) == 3) { p8 = cgetg(2, t_VEC); gel(p8, 1) = gen_0; gel(listpointstriv, i) = p8; } else { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long j; for (j = 1; j <= 2; ++j) { gel(gel(listpointstriv, i), j) = gdiv(gel(gel(listpointstriv, i), j), gpowgs(K, j)); if (low_stack(st_lim, stack_lim(btop, 1))) listpointstriv = gerepilecopy(btop, listpointstriv); } } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &p8, &listpointstriv); } } listpointstriv = concat(help, listpointstriv); if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" Trivial points on the curve = %Ps\n", listpointstriv); } /* MODI: translate points triviaux */ /* \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ */ /* Run through the Selmer group \\ */ /* \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ */ if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) { pari_printf("\n"); pari_printf(" Run through the Selmer group\n"); } listpoints = cgetg(1, t_VEC); selmer = lift(selmer); iwhile = gen_1; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long l9; GEN p10 = gen_0; /* vec */ long l11; GEN p12 = gen_0; long l13, l14; GEN p15 = gen_0, p16 = gen_0, p17 = gen_0, p18 = gen_0, p19 = gen_0, p20 = gen_0, p21 = gen_0, p22 = gen_0, p23 = gen_0, p24 = gen_0; /* vec */ while (gcmp(iwhile, shifti(gen_1, glength(selmer))) < 0) { if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf("\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" iwhile = %Ps\n", iwhile); l9 = glength(selmer); { long i; p10 = cgetg(l9+1, t_COL); for (i = 1; i <= l9; ++i) gel(p10, i) = gbittest(iwhile, i - 1); } /* the next element zc as an algebraic number modulo squares */ expo = gmul(selmer, p10); l11 = glength(LS2); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; p12 = gen_1; for (i = 1; i <= l11; ++i) { p12 = gmul(p12, gpow(gel(LS2, i), gel(expo, i), prec)); if (low_stack(st_lim, stack_lim(btop, 1))) p12 = gerepilecopy(btop, p12); } } zc = p12; if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" zc = %Ps\n", zc); liftzc = lift(zc); /* Reduction modulo squares */ if (!gequal0(redflag)) { zc = reducemodsquares(zc, gen_2, prec); liftzc = lift(zc); den = gsqr(denom(content(liftzc))); zc = gmul(zc, den); liftzc = gmul(liftzc, den); if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" zc reduced = %Ps\n", zc); } /* Does it come from a trivial point ? */ l13 = glength(listpointstriv); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; GEN p25 = gen_0; /* vec */ for (i = 1; i <= l13; ++i) { point = gcopy(gel(listpointstriv, i)); if (glength(point) == 2) if (nfissquare(member_nf(bnf), gmul(gmul(K, gsub(gel(point, 1), x)), liftzc), prec)) { if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" comes from the trivial point %Ps\n", point); p25 = cgetg(2, t_VEC); gel(p25, 1) = gcopy(point); listpoints = concat(listpoints, p25); iwhile = shifti(gen_1, gtos(gaddgs(degre(iwhile), 1))); goto label3; } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 4, &point, &p25, &listpoints, &iwhile); } } if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" does not come from a trivial point\n"); /* Construction of the quadratic form q2 */ /* Change the basis using the square factors of zc */ idealfactorzc = idealfactor(bnf, zc); gel(idealfactorzc, 2) = gmulgs(gel(idealfactorzc, 2), -1); gel(idealfactorzc, 2) = gdivent(gel(idealfactorzc, 2), gen_2); /* idealzc = idealfactorback(bnf,idealfactorzc); */ idealzc = matid(3); l14 = glength(gel(idealfactorzc, 1)); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; for (i = 1; i <= l14; ++i) { idealzc = idealmul(bnf, idealzc, idealpow0(bnf, gcoeff(idealfactorzc, i, 1), gcoeff(idealfactorzc, i, 2), 0)); if (low_stack(st_lim, stack_lim(btop, 1))) idealzc = gerepilecopy(btop, idealzc); } } { long i; p15 = cgetg(4, t_VEC); for (i = 1; i <= 3; ++i) gel(p15, i) = basistoalg(bnf, gel(idealzc, i)); } baseidealzc = p15; { long i, j; p16 = cgetg(4, t_MAT); for (j = 1; j <= 3; ++j) { gel(p16, j) = cgetg(4, t_COL); for (i = 1; i <= 3; ++i) gcoeff(p16, i, j) = gtrace(gdiv(gmul(gmul(zc, gel(baseidealzc, i)), gel(baseidealzc, j)), polprime)); } } q2 = p16; if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" q2 = %Ps\n", q2); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" q2/content(q2) = %Ps\n", gdiv(q2, content(q2))); /* Solution of the quadratic equation q2=0 */ sol = Qfsolve(gdiv(q2, content(q2)), gen_0, prec); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" sol = %Ps\n", sol); if (typ(sol) == t_INT) pari_err(user, " ell2descent_gen : WRONG ELEMENT IN THE SELMER GROUP, please report"); p17 = cgetg(4, t_COL); gel(p17, 1) = gsqr(x); gel(p17, 2) = gcopy(x); gel(p17, 3) = gen_1; /* Parametrizing the solutions of q2=0 */ param = gmul(Qfparam(q2, sol, NULL, prec), p17); param = gdiv(param, content(param)); { long i, j; p18 = cgetg(4, t_MAT); for (j = 1; j <= 3; ++j) { gel(p18, j) = cgetg(4, t_COL); for (i = 1; i <= 3; ++i) gcoeff(p18, i, j) = gtrace(gdiv(gmul(gmul(gmul(zc, gel(baseidealzc, i)), gel(baseidealzc, j)), gadd(ttheta, A)), polprime)); } } /* Construction of the quartic */ q1 = gneg(p18); pol = gmul(gmul(gtrans(param), q1), param); if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" quartic: %Ps*Y^2 = %Ps\n", K, pol); redq = redquartic(pol, prec); pol = gcopy(gel(redq, 1)); den = denom(content(gmul(K, pol))); pol = gmul(pol, gsqr(den)); if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" reduced: %Ps*Y^2 = %Ps\n", K, pol); /* Search for points on the quartic */ point = ratpoint(gmul(K, pol), LIM1, gen_1, prec); if (gequal(point, cgetg(1, t_VEC))) point = ratpoint2(gmul(K, pol), LIM3, gen_1, NULL, prec); if (gequal(point, cgetg(1, t_VEC))) { iwhile = gaddgs(iwhile, 1); continue; } if (glength(point) == 2) { p19 = cgetg(2, t_VEC); gel(p19, 1) = gen_1; point = concat(point, p19); } if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" point on the reduced quartic = %Ps\n", point); p20 = cgetg(3, t_COL); gel(p20, 1) = gcopy(gel(point, 1)); gel(p20, 2) = gcopy(gel(point, 3)); p21 = cgetg(2, t_COL); gel(p21, 1) = gdiv(gel(point, 2), den); point = concat(gmul(gel(redq, 2), p20), p21); if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" point on the quartic = %Ps\n", point); /* Construction of the point on the elliptic curve from the point on the quartic */ param = gmul(gsubst(param, gvar(x), gdiv(x, y)), gsqr(y)); param = gsubst(gsubst(param, gvar(x), gel(point, 1)), gvar(y), gel(point, 2)); param = gmul(param, gdiv(K, gel(point, 3))); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" reconstruction of the point on the curve\n"); { long i, j; p22 = cgetg(4, t_MAT); for (j = 1; j <= 3; ++j) { gel(p22, j) = cgetg(4, t_COL); for (i = 1; i <= 3; ++i) gcoeff(p22, i, j) = gtrace(gdiv(gmul(gmul(gmul(zc, gel(baseidealzc, i)), gel(baseidealzc, j)), gadd(gadd(gsqr(ttheta), gmul(A, ttheta)), B)), polprime)); } } q0 = p22; pointxx = gdiv(gmul(gmul(gtrans(param), q0), param), K); p23 = cgetg(3, t_VEC); gel(p23, 1) = gcopy(pointxx); gel(p23, 2) = sqrtrat(gdiv(gsubst(polrel, gvar(x), pointxx), K)); point2 = p23; if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) pari_printf(" point on the curve = %Ps\n", point2); p24 = cgetg(2, t_VEC); gel(p24, 1) = gcopy(point2); listpoints = concat(listpoints, p24); iwhile = shifti(gen_1, gtos(gaddgs(degre(iwhile), 1))); label3:; if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 31, &p10, &expo, &p12, &zc, &liftzc, &den, &point, &listpoints, &iwhile, &idealfactorzc, &idealzc, &p15, &baseidealzc, &p16, &q2, &sol, &p17, ¶m, &p18, &q1, &pol, &redq, &p19, &p20, &p21, &p22, &q0, &pointxx, &p23, &point2, &p24); } } /* MODI normalized blanks */ /* \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ */ /* Conclusion report \\ */ /* \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ */ rang = stoi(glength(listpoints)); if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) { pari_printf("\n"); pari_printf(" rank of found points = %ld\n", glength(listpoints)); pari_printf(" rank of the Selmer group = %ld\n", glength(selmer)); } if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) afficheselmer(rang, stoi(glength(selmer)), gen_0); if (!gequal0(gmodgs(gsubsg(glength(selmer), rang), 2))) { rang = gaddgs(rang, 1); if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) { pari_printf(" III should be a square, hence \n"); afficheselmer(rang, stoi(glength(selmer)), gen_0); } } /* Verification */ if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) pari_printf("listpoints = %Ps\n", listpoints); l6 = glength(listpoints); { pari_sp btop = avma; long i; for (i = 1; i <= l6; ++i) { if (!gequal0(gsub(gsubst(polrel, gvar(x), gel(gel(listpoints, i), 1)), gmul(K, gsqr(gel(gel(listpoints, i), 2)))))) pari_err(user, " ell2descent_gen : WRONG POINT = %Ps please report", gel(listpoints, i)); avma = btop; } } /* Reduction of the points */ listpoints = vecsort0(listpoints, NULL, 2); if ((glength(listpoints) >= 2) && !gequal0(ELLREDGENFLAG)) listpoints = ellredgen(ell, listpoints, K, prec); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" end of ell2descent_gen\n"); p7 = cgetg(4, t_VEC); gel(p7, 1) = gcopy(rang); gel(p7, 2) = stoi(glength(selmer)); gel(p7, 3) = gcopy(listpoints); p7 = gerepilecopy(ltop, p7); return p7; } void afficheselmer(GEN m1, GEN m2, GEN tors2) /* void */ { pari_sp ltop = avma; pari_printf("#E(Q)[2] = %Ps\n", shifti(gen_1, gtos(tors2))); pari_printf("#S(E/Q)[2] = %Ps\n", shifti(gen_1, gtos(m2))); if (gequal(gadd(m1, tors2), m2)) { pari_printf("#E(Q)/2E(Q) = %Ps\n", shifti(gen_1, gtos(gadd(m1, tors2)))); pari_printf("#III(E/Q)[2] = 1\n"); pari_printf("rank(E/Q) = %Ps\n", m1); } else { pari_printf("#E(Q)/2E(Q) >= %Ps\n", shifti(gen_1, gtos(gadd(m1, tors2)))); pari_printf("#III(E/Q)[2] <= %Ps\n", shifti(gen_1, gtos(gsub(gsub(m2, m1), tors2)))); pari_printf("rank(E/Q) >= %Ps\n", m1); } avma = ltop; return; } /* MODI: HC added */ GEN ellrankdebug(GEN ell, GEN lev, GEN help, long prec) { pari_sp ltop = avma; GEN p1 = gen_0; if (!help) help = cgetg(1, t_VEC); DEBUGLEVEL_ell = gcopy(lev); p1 = ellrank(ell, help, prec); gerepileall(ltop, 2, &DEBUGLEVEL_ell, &p1); return p1; } GEN ellrank(GEN ell, GEN help, long prec) { pari_sp ltop = avma; GEN urst = gen_0, urst1 = gen_0, den = gen_0, eqell = gen_0, tors2 = gen_0, bnf = gen_0, rang = gen_0, time1 = gen_0; GEN p1 = gen_0, p2 = gen_0, p3 = gen_0, p4 = gen_0, p5 = gen_0, p6 = gen_0; /* vec */ if (!help) help = cgetg(1, t_VEC); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" starting ellrank\n"); if (glength(ell) < 13) ell = smallellinit(ell); p1 = cgetg(5, t_VEC); gel(p1, 1) = gen_1; gel(p1, 2) = gen_0; gel(p1, 3) = gen_0; gel(p1, 4) = gen_0; /* kill the coefficients a1 and a3 */ urst = p1; if (!gequalgs(ell_get_a1(ell), 0) || !gequalgs(ell_get_a3(ell), 0)) { p2 = cgetg(5, t_VEC); gel(p2, 1) = gen_1; gel(p2, 2) = gen_0; gel(p2, 3) = gdivgs(gneg(ell_get_a1(ell)), 2); gel(p2, 4) = gdivgs(gneg(ell_get_a3(ell)), 2); urst1 = p2; ell = ellchangecurve(ell, urst1); urst = ellcomposeurst(urst, urst1); } /* kill denominators */ { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN p7 = gen_0; /* vec */ long l8; GEN p9 = gen_0, p10 = gen_0; /* vec */ for(;;) { p7 = cgetg(4, t_VEC); gel(p7, 1) = gcopy(ell_get_a2(ell)); gel(p7, 2) = gcopy(ell_get_a4(ell)); gel(p7, 3) = gcopy(ell_get_a6(ell)); if (!(gcmpgs(den = denom(p7), 1) > 0)) break; den = factor(den); l8 = glength(gel(den, 2)); { long i; p9 = cgetg(l8+1, t_COL); for (i = 1; i <= l8; ++i) gel(p9, i) = gen_1; } gel(den, 2) = p9; den = factorback(den); p10 = cgetg(5, t_VEC); gel(p10, 1) = ginv(den); gel(p10, 2) = gen_0; gel(p10, 3) = gen_0; gel(p10, 4) = gen_0; urst1 = p10; ell = ellchangecurve(ell, urst1); urst = ellcomposeurst(urst, urst1); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 7, &p7, &den, &p9, &p10, &urst1, &ell, &urst); } } help = ellchangepoint(help, urst); p3 = cgetg(5, t_VEC); gel(p3, 1) = gen_1; gel(p3, 2) = gcopy(ell_get_a2(ell)); gel(p3, 3) = gcopy(ell_get_a4(ell)); gel(p3, 4) = gcopy(ell_get_a6(ell)); eqell = gtopoly(p3, -1); if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) pari_printf(" Elliptic curve : Y^2 = %Ps\n", eqell); p4 = cgetg(2, t_VEC); gel(p4, 1) = gen_0; /* Choice of the algorithm depending on the 2-torsion structure */ tors2 = ellhalf(ell, p4, prec); if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) pari_printf(" E[2] = %Ps\n", tors2); if (glength(tors2) == 1) { /* case 1: 2-torsion trivial */ if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" bnfinit(%Ps)", eqell); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) gettime(); bnf = Buchall(eqell, nf_FORCE, prec); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) time1 = stoi(gettime()); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" done\n"); rang = ell2descent_gen(ell, bnf, gen_1, help, NULL, prec); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" time for bnfinit = %Ps\n", time1); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" time for the rest = %ld\n", gettime()); } else { if ((glength(tors2) == 2) || (gequal0(COMPLETE))) { /* case 2: 2-torsion >= Z/2Z */ if (!gequalgs(ell_get_a6(ell), 0)) { p5 = cgetg(5, t_VEC); gel(p5, 1) = gen_1; gel(p5, 2) = gcopy(gel(gel(tors2, 2), 1)); gel(p5, 3) = gen_0; gel(p5, 4) = gen_0; urst1 = p5; ell = ellchangecurve(ell, urst1); urst = ellcomposeurst(urst, urst1); } p6 = cgetg(5, t_VEC); gel(p6, 1) = gen_1; gel(p6, 2) = gcopy(ell_get_a2(ell)); gel(p6, 3) = gcopy(ell_get_a4(ell)); gel(p6, 4) = gcopy(ell_get_a6(ell)); eqell = gtopoly(p6, -1); if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) pari_printf(" Elliptic curve : Y^2 = %Ps\n", eqell); rang = ell2descent_viaisog(ell, help, prec); } else /* case 3: 2-torsion = Z/2Z*Z/2Z */ rang = ell2descent_complete(gel(gel(tors2, 2), 1), gel(gel(tors2, 3), 1), gel(gel(tors2, 4), 1), gen_0, prec); } gel(rang, 3) = ellchangepointinverse(gel(rang, 3), urst); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" end of ellrank\n"); rang = gerepilecopy(ltop, rang); return rang; } GEN ell2descent_complete(GEN e1, GEN e2, GEN e3, GEN help, long prec) /* vec */ { pari_sp ltop = avma; GEN ee = gen_0, d32 = gen_0, d31 = gen_0, d21 = gen_0, G1 = gen_0, G2 = gen_0, G3 = gen_0, vect1 = gen_0, vect2 = gen_0, vect3 = gen_0, selmer = gen_0, rang = gen_0, listepoints = gen_0, b1 = gen_0, b2 = gen_0, q1 = gen_0, sol1 = gen_0, param1 = gen_0, param1x = gen_0, quart = gen_0, point = gen_0, z1 = gen_0, solx = gen_0, soly = gen_0, strange = gen_0, ell = gen_0; GEN p1 = gen_0; /* vec */ long l2; GEN p3 = gen_0; /* vec */ long l4; GEN p5 = gen_0; /* vec */ GEN x = pol_x(fetch_user_var("x")); GEN p6 = gen_0, p7 = gen_0; /* vec */ /* local(sol2,q2); */ /* only if used below */ /* MODI remove sol2 q2 */ if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" Algorithm of complete 2-descent\n"); p1 = cgetg(4, t_VEC); gel(p1, 1) = gcopy(e1); gel(p1, 2) = gcopy(e2); gel(p1, 3) = gcopy(e3); /* sort the integers e1, e2, e3 in increasing order */ ee = sort(p1); e1 = gcopy(gel(ee, 1)); e2 = gcopy(gel(ee, 2)); e3 = gcopy(gel(ee, 3)); /* Computation of the groups G1 and G2 */ d32 = gsub(e3, e2); d31 = gsub(e3, e1); d21 = gsub(e2, e1); G1 = gcopy(gel(factor(gmul(d31, d21)), 1)); /* (G1 > 0) */ G2 = gcopy(gel(factor(gmul(gneg(d32), d21)), 1)); /* (G2 < 0) */ G3 = gmul(d31, d32); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" G1 = %Ps\n", G1); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" G2 = %Ps\n", G2); l2 = glength(G1); { long i; GEN p8 = gen_0; /* vec */ p3 = cgetg(l2+1, t_VEC); for (i = 1; i <= l2; ++i) { p8 = cgetg(3, t_VEC); gel(p8, 1) = gen_0; gel(p8, 2) = gen_1; gel(p3, i) = p8; } } /* Run through G1*G2 */ vect1 = p3; l4 = glength(G2); { long i; GEN p9 = gen_0; /* vec */ p5 = cgetg(l4+1, t_VEC); for (i = 1; i <= l4; ++i) { p9 = cgetg(3, t_VEC); gel(p9, 1) = gen_0; gel(p9, 2) = gen_1; gel(p5, i) = p9; } } vect2 = p5; selmer = gen_0; rang = gen_0; listepoints = cgetg(1, t_VEC); { GEN fv_data = gen_0; GEN (*fv_next)(GEN, GEN); /* func_GG */ GEN X = forvec_start(vect1, 0, &fv_data, &fv_next); /* vec */ { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long l10; GEN p11 = gen_0; long l12; for ( ; X; X = fv_next(fv_data, X)) { l10 = glength(G1); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; p11 = gen_1; for (i = 1; i <= l10; ++i) { p11 = gmul(p11, gpow(gel(G1, i), gel(X, i), prec)); if (low_stack(st_lim, stack_lim(btop, 1))) p11 = gerepilecopy(btop, p11); } } b1 = p11; /* b1*b2*b3 must be a square, where b3 is a divisor of d32*d31 */ vect3 = gcopy(vect2); l12 = glength(G2); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; GEN p13 = gen_0; /* vec */ for (i = 2; i <= l12; ++i) { if (!gequalgs(gmod(G3, gel(G2, i)), 0)) { p13 = cgetg(3, t_VEC); gel(p13, 1) = gen_1; gel(p13, 2) = gen_1; gel(vect3, i) = gmulgs(p13, ggval(b1, gel(G2, i))); } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &p13, &vect3); } } { GEN fv_data = gen_0; GEN (*fv_next)(GEN, GEN); /* func_GG */ GEN Y = forvec_start(vect3, 0, &fv_data, &fv_next); /* vec */ { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long l14; GEN p15 = gen_0; GEN p16 = gen_0, p17 = gen_0, p18 = gen_0, p19 = gen_0, p20 = gen_0, p21 = gen_0, p22 = gen_0; /* vec */ long l23; GEN p24 = gen_0, p25 = gen_0, p26 = gen_0, p27 = gen_0, p28 = gen_0; /* vec */ for ( ; Y; Y = fv_next(fv_data, Y)) { l14 = glength(G2); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; p15 = gen_1; for (i = 1; i <= l14; ++i) { p15 = gmul(p15, gpow(gel(G2, i), gel(Y, i), prec)); if (low_stack(st_lim, stack_lim(btop, 1))) p15 = gerepilecopy(btop, p15); } } b2 = p15; if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) { p16 = cgetg(3, t_VEC); gel(p16, 1) = gcopy(b1); gel(p16, 2) = gcopy(b2); pari_printf(" [b1,b2] = %Ps\n", lift(p16)); } /* Trivial points coming from the 2-torsion */ if (gequal1(b1) && gequal1(b2)) { if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" trivial point [0]\n"); selmer = gaddgs(selmer, 1); rang = gaddgs(rang, 1); continue; } if (!gequal0(gissquare(gmul(gneg(d21), b2))) && !gequal0(gissquare(gmul(gmul(d31, d21), b1)))) { if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" trivial point [e1,0]\n"); selmer = gaddgs(selmer, 1); rang = gaddgs(rang, 1); p17 = cgetg(2, t_VEC); p18 = cgetg(3, t_VEC); gel(p18, 1) = gcopy(e1); gel(p18, 2) = gen_0; gel(p17, 1) = p18; listepoints = concat(listepoints, p17); continue; } if (!gequal0(gissquare(gmul(d21, b1))) && !gequal0(gissquare(gmul(gmul(gneg(d32), d21), b2)))) { if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" trivial point [e2,0]\n"); selmer = gaddgs(selmer, 1); rang = gaddgs(rang, 1); p19 = cgetg(2, t_VEC); p20 = cgetg(3, t_VEC); gel(p20, 1) = gcopy(e2); gel(p20, 2) = gen_0; gel(p19, 1) = p20; listepoints = concat(listepoints, p19); continue; } if (!gequal0(gissquare(gmul(d31, b1))) && !gequal0(gissquare(gmul(d32, b2)))) { if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" trivial point [e3,0]\n"); selmer = gaddgs(selmer, 1); rang = gaddgs(rang, 1); p21 = cgetg(2, t_VEC); p22 = cgetg(3, t_VEC); gel(p22, 1) = gcopy(e3); gel(p22, 2) = gen_0; gel(p21, 1) = p22; listepoints = concat(listepoints, p21); continue; } /* Trivial points coming from help */ l23 = glength(help); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; GEN p29 = gen_0; /* vec */ for (i = 1; i <= l23; ++i) { if ((glength(gel(help, i)) != 2) || gequal0(gel(gel(help, i), 2))) continue; if (!gequal0(gissquare(gmul(b1, gsub(gel(gel(help, i), 1), e1)))) && !gequal0(gissquare(gmul(b2, gsub(gel(gel(help, i), 1), e2))))) { if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" trivial point from help %Ps\n", gel(help, i)); selmer = gaddgs(selmer, 1); rang = gaddgs(rang, 1); p29 = cgetg(2, t_VEC); gel(p29, 1) = gcopy(gel(help, i)); listepoints = concat(listepoints, p29); goto label4; } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 4, &selmer, &rang, &p29, &listepoints); } } p24 = cgetg(4, t_VEC); gel(p24, 1) = gcopy(b1); gel(p24, 2) = gneg(b2); gel(p24, 3) = gneg(d21); /* If one can solve 2 quadratic equations */ /* (1) q1 : b1*z1^2-b2*z2^2 = e2-e1 */ /* (2) q2 : b1*z1^2-b1*b2*z3^2 = e3-e1 */ /* then (x,y) = (b1*z1^2+e1,b1*b2*z1*z2*z3) is a point on E */ /* we also have */ /* (3) q3 = q1-q2 : b1*b2*z3^2-b2*z2^2=e2-e3 */ /* Solution of the q1 */ q1 = diagonal(p24); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" q1 = %Ps\n", q1); sol1 = Qfsolve(q1, gen_0, prec); if (typ(sol1) == t_INT) { if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" q1 not ELS at %Ps\n", sol1); continue; } if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" solution of q1 = %Ps\n", sol1); param1 = Qfparam(q1, sol1, gen_1, prec); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" parametrization of q1 = %Ps\n", param1); p25 = cgetg(4, t_COL); gel(p25, 1) = gsqr(x); gel(p25, 2) = gcopy(x); gel(p25, 3) = gen_1; param1x = gmul(param1, p25); /* Solution of the q2 */ /* only useful to detect local non solubility */ /* q2 = matdiagonal([b1,-b1*b2,-d31]); */ /*if( DEBUGLEVEL_ell >= 3, print(" q2 = ",q2)); */ /* sol2 = Qfsolve(q2); */ /* if( type(sol2) == "t_INT", */ /*if( DEBUGLEVEL_ell >= 3, print(" q2 not ELS at ",sol2)); */ /* next); */ /* Construction of the quartic */ quart = gmul(gmul(b1, b2), gsub(gmul(b1, gsqr(gel(param1x, 1))), gmul(d31, gsqr(gel(param1x, 3))))); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" quart = %Ps\n", quart); /* Local solubility of the quartic */ if (!locallysoluble(quart, prec)) { if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" quartic not ELS \n"); continue; } if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" y^2 = %Ps\n", quart); selmer = gaddgs(selmer, 1); /* Search for points on the quartic */ point = ratpoint2(quart, LIM3, gen_1, NULL, prec); if (!gequal(point, cgetg(1, t_VEC))) { if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" point found on the quartic !!\n"); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" %Ps\n", point); if (glength(point) == 2) z1 = gdiv(gsubst(gel(param1x, 1), gvar(x), gel(point, 1)), gsubst(gel(param1x, 3), gvar(x), gel(point, 1))); else z1 = gdiv(gcoeff(param1, 1, 1), gcoeff(param1, 3, 1)); solx = gadd(gmul(b1, gsqr(z1)), e1); soly = sqrtrat(gmul(gmul(gsub(solx, e1), gsub(solx, e2)), gsub(solx, e3))); p26 = cgetg(2, t_VEC); p27 = cgetg(3, t_VEC); gel(p27, 1) = gcopy(solx); gel(p27, 2) = gcopy(soly); gel(p26, 1) = p27; listepoints = concat(listepoints, p26); if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) { p28 = cgetg(3, t_VEC); gel(p28, 1) = gcopy(solx); gel(p28, 2) = gcopy(soly); pari_printf(" point on the elliptic curve = %Ps\n", p28); } rang = gaddgs(rang, 1); } else if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" no point found on the quartic\n"); label4:; if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 27, &p15, &b2, &p16, &selmer, &rang, &p18, &p17, &listepoints, &p20, &p19, &p22, &p21, &p24, &q1, &sol1, ¶m1, &p25, ¶m1x, &quart, &point, &z1, &solx, &soly, &p27, &p26, &p28, &Y); } } } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 17, &p11, &b1, &vect3, &b2, &selmer, &rang, &listepoints, &q1, &sol1, ¶m1, ¶m1x, &quart, &point, &z1, &solx, &soly, &X); } } } /* end */ if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) pari_printf("#S^(2) = %Ps\n", selmer); if (gcmp(rang, gdivgs(selmer, 2)) > 0) rang = selmer; if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) { strange = stoi(!gequal(rang, selmer)); if (!gequal0(strange)) pari_printf("#E[K]/2E[K]>= %Ps\n", rang); else pari_printf("#E[K]/2E[K] = %Ps\n", rang); pari_printf("#E[2] = 4\n"); } rang = gsubgs(gceil(gdiv(glog(rang, prec), glog(gen_2, prec))), 2); selmer = stoi(ggval(selmer, gen_2)); if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) { if (!gequal0(strange)) pari_printf("%Ps >= rank >= %Ps\n", gsubgs(selmer, 2), rang); else pari_printf("rank = %Ps\n", rang); if (!gequal0(rang)) pari_printf("points = %Ps\n", listepoints); } p6 = cgetg(6, t_VEC); gel(p6, 1) = gen_0; gel(p6, 2) = gneg(gadd(gadd(e1, e2), e3)); gel(p6, 3) = gen_0; gel(p6, 4) = gadd(gadd(gmul(e1, e2), gmul(e2, e3)), gmul(e3, e1)); gel(p6, 5) = gmul(gmul(gneg(e1), e2), e3); ell = smallellinit(p6); listepoints = vecsort0(listepoints, NULL, 2); if (!gequal0(ELLREDGENFLAG)) listepoints = ellredgen(ell, listepoints, NULL, prec); listepoints = concat(ellsort(gel(elltorseven(ell, prec), 3), prec), listepoints); p7 = cgetg(4, t_VEC); gel(p7, 1) = gcopy(rang); gel(p7, 2) = gcopy(selmer); gel(p7, 3) = gcopy(listepoints); p7 = gerepilecopy(ltop, p7); return p7; } GEN ellcount(GEN c, GEN d, GEN KS2gen, GEN listpointstriv, long prec) /* vec */ { pari_sp ltop = avma; GEN found = gen_0, listgen = gen_0, listpointscount = gen_0, m1 = gen_0, m2 = gen_0, lastloc = gen_0, mask = gen_0, i = gen_0, d1 = gen_0, iaux = gen_0, j = gen_0, triv = gen_0, pol = gen_0, point = gen_0, qf = gen_0, solqf = gen_0, para = gen_0, point1 = gen_0, v = gen_0; GEN p1 = gen_0; /* vec */ GEN x = pol_x(fetch_user_var("x")), t = pol_x(fetch_user_var("t")); long l2; GEN p3 = gen_0, p4 = gen_0; /* vec */ if (!listpointstriv) listpointstriv = cgetg(1, t_VEC); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) { p1 = cgetg(3, t_VEC); gel(p1, 1) = gcopy(c); gel(p1, 2) = gcopy(d); pari_printf(" starting ellcount %Ps\n", p1); } if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" KS2gen = %Ps\n", KS2gen); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" listpointstriv = %Ps\n", listpointstriv); found = gen_0; listgen = gcopy(KS2gen); listpointscount = cgetg(1, t_VEC); m1 = m2 = gen_0; lastloc = gen_m1; mask = shifti(gen_1, glength(KS2gen)); i = gen_1; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long l5; GEN p6 = gen_0, p7 = gen_0, p8 = gen_0, p9 = gen_0, p10 = gen_0, p11 = gen_0, p12 = gen_0, p13 = gen_0, p14 = gen_0, p15 = gen_0; /* vec */ while (gcmp(i, mask) < 0) { d1 = gen_1; iaux = i; j = gen_1; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); while (!gequal0(iaux)) { if (!gequal0(gmodgs(iaux, 2))) d1 = gmul(d1, gel(listgen, gtos(j))); iaux = gshift(iaux, -1); j = gaddgs(j, 1); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 3, &d1, &iaux, &j); } } if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" d1 = %Ps\n", d1); triv = gen_0; l5 = glength(listpointstriv); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long j; GEN p16 = gen_0; /* vec */ for (j = 1; j <= l5; ++j) { if (!gequal0(gel(gel(listpointstriv, j), 1)) && !gequal0(gissquare(gmul(d1, gel(gel(listpointstriv, j), 1))))) { p16 = cgetg(2, t_VEC); gel(p16, 1) = gcopy(gel(listpointstriv, j)); listpointscount = concat(listpointscount, p16); if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" trivial point\n"); triv = gen_1; m1 = gaddgs(m1, 1); if (gcmp(degre(i), lastloc) > 0) m2 = gaddgs(m2, 1); found = gen_1; lastloc = gen_m1; break; } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 7, &p16, &listpointscount, &triv, &m1, &m2, &found, &lastloc); } } if (gequal0(triv)) { p6 = cgetg(6, t_VEC); gel(p6, 1) = gcopy(d1); gel(p6, 2) = gen_0; gel(p6, 3) = gcopy(c); gel(p6, 4) = gen_0; gel(p6, 5) = gdiv(d, d1); pol = gtopoly(p6, -1); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" quartic = y^2 = %Ps\n", pol); point = ratpoint(pol, LIM1, gen_1, prec); if (!gequal(point, cgetg(1, t_VEC))) { if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" point on the quartic\n"); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf("%Ps\n", point); m1 = gaddgs(m1, 1); p7 = cgetg(2, t_VEC); gel(p7, 1) = gmul(gmul(d1, gel(point, 1)), point); listpointscount = concat(listpointscount, p7); if (gcmp(degre(i), lastloc) > 0) m2 = gaddgs(m2, 1); found = gen_1; lastloc = gen_m1; } else if (locallysoluble(pol, prec)) { if (gcmp(degre(i), lastloc) > 0) { m2 = gaddgs(m2, 1); lastloc = degre(i); } p8 = cgetg(4, t_MAT); gel(p8, 1) = cgetg(4, t_COL); gel(p8, 2) = cgetg(4, t_COL); gel(p8, 3) = cgetg(4, t_COL); gcoeff(p8, 1, 1) = gcopy(d1); gcoeff(p8, 1, 2) = gdivgs(c, 2); gcoeff(p8, 1, 3) = gen_0; gcoeff(p8, 2, 1) = gdivgs(c, 2); gcoeff(p8, 2, 2) = gdiv(d, d1); gcoeff(p8, 2, 3) = gen_0; gcoeff(p8, 3, 1) = gen_0; gcoeff(p8, 3, 2) = gen_0; gcoeff(p8, 3, 3) = gen_m1; /* point = ratpoint2(pol,LIM3,1); */ /*\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ */ /* Instead of solving directly y^2 = d1*x^4+c*x^2+d/d1, */ /* we solve first y^2 = d1*X^2+c*X+d/d1, then solve the quartic X = x^2 */ /* which gives a new quartic */ qf = gmulsg(2, p8); solqf = Qfsolve(qf, gen_0, prec); p9 = cgetg(4, t_COL); gel(p9, 1) = gsqr(x); gel(p9, 2) = gcopy(x); gel(p9, 3) = gen_1; para = gmul(Qfparam(qf, solqf, gen_2, prec), p9); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) { p10 = cgetg(4, t_VEC); gel(p10, 1) = gcopy(d1); gel(p10, 2) = gcopy(c); gel(p10, 3) = gdiv(d, d1); pari_printf(" the conic y^2 = %Ps\n", gtopoly(p10, -1)); p11 = cgetg(3, t_VEC); gel(p11, 1) = gdiv(gel(para, 1), gel(para, 2)); gel(p11, 2) = gdiv(gel(para, 3), gel(para, 2)); pari_printf(" is parametrized by [x,y] = %Ps\n", gsubst(p11, gvar(x), t)); } point1 = ratpoint2(gmul(gel(para, 1), gel(para, 2)), LIM3, gen_1, NULL, prec); if (!gequal(point1, cgetg(1, t_VEC))) { if (glength(point1) == 2) para = gsubst(para, gvar(x), gel(point1, 1)); else { p12 = cgetg(4, t_VEC); gel(p12, 1) = gen_1; gel(p12, 2) = gdiv(gel(point1, 2), gsqr(gel(point1, 1))); gel(p12, 3) = gen_0; point1 = p12; { long ii; p13 = cgetg(4, t_VEC); for (ii = 1; ii <= 3; ++ii) gel(p13, ii) = polcoeff0(gel(para, ii), 2, -1); } para = p13; } p14 = cgetg(3, t_VEC); gel(p14, 1) = gdiv(gel(point1, 2), gel(para, 2)); gel(p14, 2) = gdiv(gel(para, 3), gel(para, 2)); point = p14; } else point = cgetg(1, t_VEC); /*\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ */ if (!gequal(point, cgetg(1, t_VEC))) { if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" point on the quartic\n"); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf("%Ps\n", point); m1 = gaddgs(m1, 1); p15 = cgetg(2, t_VEC); gel(p15, 1) = gmul(gmul(d1, gel(point, 1)), point); listpointscount = concat(listpointscount, p15); if (gcmp(degre(i), lastloc) > 0) m2 = gaddgs(m2, 1); found = gen_1; lastloc = gen_m1; } else if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" no point found on the quartic\n"); } } if (!gequal0(found)) { found = gen_0; v = gen_0; iaux = gshift(i, -1); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); while (!gequal0(iaux)) { iaux = gshift(iaux, -1); v = gaddgs(v, 1); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &iaux, &v); } } mask = gshift(mask, -1); listgen = extract0(listgen, subis(subii(shifti(gen_1, glength(listgen)), shifti(gen_1, gtos(v))), 1), NULL); i = shifti(gen_1, gtos(v)); } else i = gaddgs(i, 1); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 29, &d1, &iaux, &j, &triv, &listpointscount, &m1, &m2, &found, &lastloc, &p6, &pol, &point, &p7, &p8, &qf, &solqf, &p9, ¶, &p10, &p11, &point1, &p12, &p13, &p14, &p15, &v, &mask, &listgen, &i); } } l2 = glength(listpointscount); { pari_sp btop = avma; long i; for (i = 1; i <= l2; ++i) { if (glength(gel(listpointscount, i)) > 1) if (!gequalgs(gsub(gsubst(gadd(gadd(gpowgs(x, 3), gmul(c, gsqr(x))), gmul(d, x)), gvar(x), gel(gel(listpointscount, i), 1)), gsqr(gel(gel(listpointscount, i), 2))), 0)) pari_err(user, " ellcount : WRONG POINT, please report %ld", i); avma = btop; } } if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" end of ellcount\n"); p3 = cgetg(3, t_VEC); gel(p3, 1) = gcopy(listpointscount); p4 = cgetg(3, t_VEC); gel(p4, 1) = gcopy(m1); gel(p4, 2) = gcopy(m2); gel(p3, 2) = p4; p3 = gerepilecopy(ltop, p3); return p3; } GEN ell2descent_viaisog(GEN ell, GEN help, long prec) /* vec */ { pari_sp ltop = avma; GEN P = gen_0, Pfact = gen_0, tors = gen_0, listpointstriv = gen_0, KS2prod = gen_0, KS2gen = gen_0, listpoints = gen_0, pointgen = gen_0, n1 = gen_0, n2 = gen_0, certain = gen_0, apinit = gen_0, bpinit = gen_0, np1 = gen_0, np2 = gen_0, listpoints2 = gen_0, aux1 = gen_0, aux2 = gen_0, certainp = gen_0, rang = gen_0, strange = gen_0; GEN p1 = gen_0; /* vec */ GEN x = pol_x(fetch_user_var("x")); GEN p2 = gen_0, p3 = gen_0; /* vec */ long l4; GEN p5 = gen_0; /* vec */ long l6; GEN p7 = gen_0; /* vec */ if (!help) help = cgetg(1, t_VEC); if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" Algorithm of 2-descent via isogenies\n"); if (glength(ell) < 13) ell = smallellinit(ell); if (gequal0(member_disc(ell))) pari_err(user, " ell2descent_viaisog : singular curve !!"); if ((!gequalgs(ell_get_a1(ell), 0) || !gequalgs(ell_get_a3(ell), 0)) || !gequalgs(ell_get_a6(ell), 0)) pari_err(user, " ell2descent_viaisog : the curve is not on the form [0,a,0,b,0]"); if ((gcmpgs(denom(ell_get_a2(ell)), 1) > 0) || (gcmpgs(denom(ell_get_a4(ell)), 1) > 0)) pari_err(user, " ell2descent_viaisog : non-integral coefficients"); p1 = cgetg(4, t_VEC); gel(p1, 1) = gen_1; gel(p1, 2) = gcopy(ell_get_a2(ell)); gel(p1, 3) = gcopy(ell_get_a4(ell)); /* */ /* Working with the initial curve */ /* */ /* Construction of trivial points : torsion */ P = gtopoly(p1, -1); Pfact = gcopy(gel(factor(P), 1)); tors = stoi(glength(Pfact)); listpointstriv = concat(help, gel(elltorseven(ell, prec), 3)); /* Construction of trivial points : small naive height */ if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" Search for trivial points on the curve\n"); P = gmul(P, x); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" Y^2 = %Ps\n", P); listpointstriv = concat(listpointstriv, ratpoint(P, LIMTRIV, gen_0, prec)); if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) { pari_printf(" trivial points on E(Q) = %Ps\n", listpointstriv); pari_printf("\n"); } KS2prod = gneg(gabs(ell_get_a4(ell), prec)); if (gcmpgs(gsub(gsqr(ell_get_a2(ell)), gmulsg(4, ell_get_a4(ell))), 0) < 0) KS2prod = gmulgs(KS2prod, -1); KS2gen = gcopy(gel(factor(KS2prod), 1)); if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) { pari_printf(" #K(b,2)gen = %ld\n", glength(KS2gen)); pari_printf(" K(b,2)gen = %Ps\n", KS2gen); } listpoints = ellcount(ell_get_a2(ell), ell_get_a4(ell), KS2gen, listpointstriv, prec); pointgen = gcopy(gel(listpoints, 1)); if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) { pari_printf(" points on E(Q) = %Ps\n", pointgen); pari_printf("\n"); } n1 = gcopy(gel(gel(listpoints, 2), 1)); n2 = gcopy(gel(gel(listpoints, 2), 2)); certain = stoi(gequal(n1, n2)); if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) { if (!gequal0(certain)) { pari_printf("[E(Q):phi'(E'(Q))] = %Ps\n", shifti(gen_1, gtos(n1))); pari_printf("#S^(phi')(E'/Q) = %Ps\n", shifti(gen_1, gtos(n2))); pari_printf("#III(E'/Q)[phi'] = 1\n"); pari_printf("\n"); } else { pari_printf("[E(Q):phi'(E'(Q))] >= %Ps\n", shifti(gen_1, gtos(n1))); pari_printf("#S^(phi')(E'/Q) = %Ps\n", shifti(gen_1, gtos(n2))); pari_printf("#III(E'/Q)[phi'] <= %Ps\n", shifti(gen_1, gtos(gsub(n2, n1)))); pari_printf("\n"); } } /* */ /* Working with the isogeneous curve */ /* */ apinit = gmulsg(-2, ell_get_a2(ell)); bpinit = gsub(gsqr(ell_get_a2(ell)), gmulsg(4, ell_get_a4(ell))); KS2prod = gneg(gabs(bpinit, prec)); if (gcmpgs(ell_get_a4(ell), 0) < 0) KS2prod = gmulgs(KS2prod, -1); KS2gen = gcopy(gel(factor(KS2prod), 1)); if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) { pari_printf(" #K(a^2-4b,2)gen = %ld\n", glength(KS2gen)); pari_printf(" K(a^2-4b,2)gen = %Ps\n", KS2gen); } p2 = cgetg(4, t_VEC); gel(p2, 1) = gen_1; gel(p2, 2) = gcopy(apinit); gel(p2, 3) = gcopy(bpinit); /* Construction of trivial points : torsion */ P = gtopoly(p2, -1); p3 = cgetg(6, t_VEC); gel(p3, 1) = gen_0; gel(p3, 2) = gcopy(apinit); gel(p3, 3) = gen_0; gel(p3, 4) = gcopy(bpinit); gel(p3, 5) = gen_0; listpointstriv = gcopy(gel(elltorseven(p3, prec), 3)); /* Construction of trivial points : small naive height */ if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" Search for trivial points on the curve\n"); P = gmul(P, x); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" Y^2 = %Ps\n", P); listpointstriv = concat(listpointstriv, ratpoint(P, LIMTRIV, gen_0, prec)); if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) { pari_printf(" trivial points on E'(Q) = %Ps\n", listpointstriv); pari_printf("\n"); } listpoints = ellcount(apinit, bpinit, KS2gen, listpointstriv, prec); if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) pari_printf(" points on E'(Q) = %Ps\n", gel(listpoints, 1)); np1 = gcopy(gel(gel(listpoints, 2), 1)); np2 = gcopy(gel(gel(listpoints, 2), 2)); l4 = glength(gel(listpoints, 1)); { long i; p5 = cgetg(l4+1, t_VEC); for (i = 1; i <= l4; ++i) gel(p5, i) = gen_0; } listpoints2 = p5; l6 = glength(gel(listpoints, 1)); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; GEN p8 = gen_0; /* vec */ for (i = 1; i <= l6; ++i) { p8 = cgetg(3, t_VEC); gel(p8, 1) = gen_0; gel(p8, 2) = gen_0; gel(listpoints2, i) = p8; aux1 = gsqr(gel(gel(gel(listpoints, 1), i), 1)); if (!gequalgs(aux1, 0)) { aux2 = gcopy(gel(gel(gel(listpoints, 1), i), 2)); gel(gel(listpoints2, i), 1) = gdivgs(gdiv(gsqr(aux2), aux1), 4); gel(gel(listpoints2, i), 2) = gdivgs(gdiv(gmul(aux2, gsub(bpinit, aux1)), aux1), 8); } else gel(listpoints2, i) = gcopy(gel(gel(listpoints, 1), i)); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 4, &p8, &listpoints2, &aux1, &aux2); } } if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) { pari_printf(" points on E(Q) = %Ps\n", listpoints2); pari_printf("\n"); } pointgen = concat(pointgen, listpoints2); certainp = stoi(gequal(np1, np2)); if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) { if (!gequal0(certainp)) { pari_printf("[E'(Q):phi(E(Q))] = %Ps\n", shifti(gen_1, gtos(np1))); pari_printf("#S^(phi)(E/Q) = %Ps\n", shifti(gen_1, gtos(np2))); pari_printf("#III(E/Q)[phi] = 1\n"); pari_printf("\n"); } else { pari_printf("[E'(Q):phi(E(Q))] >= %Ps\n", shifti(gen_1, gtos(np1))); pari_printf("#S^(phi)(E/Q) = %Ps\n", shifti(gen_1, gtos(np2))); pari_printf("#III(E/Q)[phi] <= %Ps\n", shifti(gen_1, gtos(gsub(np2, np1)))); pari_printf("\n"); } if ((gequal0(certain)) && (gcmp(np2, np1) > 0)) pari_printf("%Ps <= ", shifti(gen_1, gtos(gsub(np2, np1)))); pari_printf("#III(E/Q)[2] "); if (!gequal0(certain) && !gequal0(certainp)) pari_printf(" "); else pari_printf("<"); pari_printf("= %Ps\n", shifti(gen_1, gtos(gsub(gsub(gadd(n2, np2), n1), np1)))); pari_printf("#E(Q)[2] = %Ps\n", shifti(gen_1, gtos(tors))); } rang = gsubgs(gadd(n1, np1), 2); if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) { if (!gequal0(certain) && !gequal0(certainp)) { pari_printf("#E(Q)/2E(Q) = %Ps\n", shifti(gen_1, gtos(gadd(rang, tors)))); pari_printf("rank = %Ps\n", rang); pari_printf("\n"); } else { pari_printf("#E(Q)/2E(Q) >= %Ps\n", shifti(gen_1, gtos(gadd(rang, tors)))); pari_printf("\n"); pari_printf("%Ps <= rank <= %Ps\n", rang, gsubgs(gadd(n2, np2), 2)); pari_printf("\n"); } } strange = gmodgs(gsub(gsub(gadd(n2, np2), n1), np1), 2); if (!gequal0(strange)) { if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) { pari_printf(" !!! III should be a square !!!\n"); pari_printf("hence\n"); } if (!gequal0(certain)) { np1 = gaddgs(np1, 1); certainp = stoi(gequal(np1, np2)); if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) { if (!gequal0(certainp)) { pari_printf("[E'(Q):phi(E(Q))] = %Ps\n", shifti(gen_1, gtos(np1))); pari_printf("#S^(phi)(E/Q) = %Ps\n", shifti(gen_1, gtos(np2))); pari_printf("#III(E/Q)[phi] = 1\n"); pari_printf("\n"); } else { pari_printf("[E'(Q):phi(E(Q))] >= %Ps\n", shifti(gen_1, gtos(np1))); pari_printf("#S^(phi)(E/Q) = %Ps\n", shifti(gen_1, gtos(np2))); pari_printf("#III(E/Q)[phi] <= %Ps\n", shifti(gen_1, gtos(gsub(np2, np1)))); pari_printf("\n"); } } } else { if (!gequal0(certainp)) { n1 = gaddgs(n1, 1); certain = stoi(gequal(n1, n2)); if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) { if (!gequal0(certain)) { pari_printf("[E(Q):phi'(E'(Q))] = %Ps\n", shifti(gen_1, gtos(n1))); pari_printf("#S^(phi')(E'/Q) = %Ps\n", shifti(gen_1, gtos(n2))); pari_printf("#III(E'/Q)[phi'] = 1\n"); pari_printf("\n"); } else { pari_printf("[E(Q):phi'(E'(Q))] >= %Ps\n", shifti(gen_1, gtos(n1))); pari_printf("#S^(phi')(E'/Q) = %Ps\n", shifti(gen_1, gtos(n2))); pari_printf("#III(E'/Q)[phi'] <= %Ps\n", shifti(gen_1, gtos(gsub(n2, n1)))); pari_printf("\n"); } } } else n1 = gaddgs(n1, 1); } if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) { pari_printf("#S^(2)(E/Q) = %Ps\n", shifti(gen_1, gtos(gsubgs(gadd(n2, np2), 1)))); if ((gequal0(certain)) && (gcmp(np2, np1) > 0)) pari_printf(" %Ps <= ", shifti(gen_1, gtos(gsub(np2, np1)))); pari_printf("#III(E/Q)[2] "); if (!gequal0(certain) && !gequal0(certainp)) pari_printf(" "); else pari_printf("<"); pari_printf("= %Ps\n", shifti(gen_1, gtos(gsub(gsub(gadd(n2, np2), n1), np1)))); pari_printf("#E(Q)[2] = %Ps\n", shifti(gen_1, gtos(tors))); } rang = gsubgs(gadd(n1, np1), 2); if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) { if (!gequal0(certain) && !gequal0(certainp)) { pari_printf("#E(Q)/2E(Q) = %Ps\n", shifti(gen_1, gtos(gadd(rang, tors)))); pari_printf("\n"); pari_printf("rank = %Ps\n", rang); pari_printf("\n"); } else { pari_printf("#E(Q)/2E(Q) >= %Ps\n", shifti(gen_1, gtos(gadd(rang, tors)))); pari_printf("\n"); pari_printf("%Ps <= rank <= %Ps\n", rang, gsubgs(gadd(n2, np2), 2)); pari_printf("\n"); } } } /* end of strange */ pointgen = vecsort0(pointgen, NULL, 2); if (!gequal0(ELLREDGENFLAG)) pointgen = ellredgen(ell, pointgen, NULL, prec); pointgen = concat(ellsort(gel(elltorseven(ell, prec), 3), prec), pointgen); if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) pari_printf("points = %Ps\n", pointgen); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" end of ell2descent_viaisog\n"); p7 = cgetg(4, t_VEC); gel(p7, 1) = gcopy(rang); gel(p7, 2) = gadd(gsubgs(gadd(n2, np2), 2), tors); gel(p7, 3) = gcopy(pointgen); p7 = gerepilecopy(ltop, p7); return p7; } long nfsign_s(GEN nf, GEN a, GEN i, long prec) { pari_sp ltop = avma; GEN nf_roots = gen_0, ay = gen_0, def = gen_0; long l1, l2; if (gequal0(a)) { avma = ltop; return 0; } a = lift(a); if (typ(a) != t_POL) { l1 = gsigne(a); avma = ltop; return l1; } nf_roots = gcopy(member_roots(nf)); def = stoi(getrealprecision()); ay = gen_0; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); while (gequal0(ay) || (cmpis(precision0(ay, 0), 10) < 0)) { ay = gsubst(a, gvar(gpolvar(a)), gel(nf_roots, gtos(i))); if (gequal0(ay) || (cmpis(precision0(ay, 0), 10) < 0)) { if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" **** Warning: doubling the real precision in nfsign_s **** %ld\n", 2*getrealprecision()); setrealprecision(2*getrealprecision(), &prec); nf_roots = greal(roots0(member_pol(nf), 0, prec)); } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &ay, &nf_roots); } } setrealprecision(gtos(def), &prec); l2 = gsigne(ay); avma = ltop; return l2; } GEN nfpolratroots(GEN nf, GEN pol) { pari_sp ltop = avma; GEN f = gen_0, ans = gen_0; long l1; f = gcopy(gel(nffactor(nf, lift(pol)), 1)); ans = cgetg(1, t_VEC); l1 = glength(f); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long j; GEN p2 = gen_0; /* vec */ for (j = 1; j <= l1; ++j) { if (degree(gel(f, j)) == 1) { p2 = cgetg(2, t_VEC); gel(p2, 1) = gdiv(gneg(polcoeff0(gel(f, j), 0, -1)), polcoeff0(gel(f, j), 1, -1)); ans = concat(ans, p2); } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &p2, &ans); } } ans = gerepilecopy(ltop, ans); return ans; } GEN nfmodid2(GEN nf, GEN a, GEN ideal) { pari_sp ltop = avma; GEN p1 = gen_0, p2 = gen_0; if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("entree dans nfmodid2\n"); /* ideal doit etre sous la forme primedec */ if ((lg(member_zk(nf))-1) == 1) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nfmodid2\n"); p1 = gmul(a, gmodulsg(1, member_p(ideal))); p1 = gerepilecopy(ltop, p1); return p1; } a = mynfeltmod(nf, a, basistoalg(nf, gel(ideal, 2))); if (gequal1(ggcd(denom(content(lift(a))), member_p(ideal)))) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nfmodid2\n"); p2 = gmul(a, gmodulsg(1, member_p(ideal))); p2 = gerepilecopy(ltop, p2); return p2; } if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nfmodid2\n"); a = gerepilecopy(ltop, a); return a; } GEN nfhilb2(GEN nf, GEN a, GEN b, GEN p, long prec) { pari_sp ltop = avma; GEN res = gen_0, x = pol_x(fetch_user_var("x")); if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("entree dans nfhilb2\n"); if (nfqpsoluble(nf, gadd(gmul(a, gsqr(x)), b), initp(nf, p, prec), prec)) res = gen_1; else res = gen_m1; if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nfhilb2\n"); res = gerepilecopy(ltop, res); return res; } GEN mynfhilbertp(GEN nf, GEN a, GEN b, GEN p, long prec) { pari_sp ltop = avma; GEN alpha = gen_0, beta = gen_0, sig = gen_0, aux = gen_0, aux2 = gen_0, rep = gen_0, p1 = gen_0; GEN p2 = gen_0; /* vec */ if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("entree dans mynfhilbertp %Ps\n", p); if (gequal0(a) || gequal0(b)) pari_printf("0 argument in mynfhilbertp\n"); if (cmpis(member_p(p), 2) == 0) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de mynfhilbertp\n"); p1 = nfhilb2(nf, a, b, p, prec); p1 = gerepilecopy(ltop, p1); return p1; } if (typ(a) != t_POLMOD) a = gmodulo(a, member_pol(nf)); if (typ(b) != t_POLMOD) b = gmodulo(b, member_pol(nf)); alpha = stoi(idealval(nf, a, p)); beta = stoi(idealval(nf, b, p)); if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) { p2 = cgetg(3, t_VEC); gel(p2, 1) = gcopy(alpha); gel(p2, 2) = gcopy(beta); pari_printf("[alpha,beta] = %Ps\n", p2); } if (gequal0(gmodgs(alpha, 2)) && gequal0(gmodgs(beta, 2))) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de mynfhilbertp\n"); avma = ltop; return gen_1; } aux2 = gdiventgs(idealnorm(nf, p), 2); if ((!gequal0(gmodgs(alpha, 2)) && !gequal0(gmodgs(beta, 2))) && !gequal0(gmodgs(aux2, 2))) sig = gen_1; else sig = gen_m1; if (!gequal0(beta)) aux = nfmodid2(nf, gdiv(gpow(a, beta, prec), gpow(b, alpha, prec)), p); else aux = nfmodid2(nf, gpow(b, alpha, prec), p); aux = gadd(gpow(aux, aux2, prec), sig); aux = lift(lift(aux)); if (gequal0(aux)) rep = gen_1; else rep = stoi(idealval(nf, aux, p) >= 1); if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de mynfhilbertp\n"); if (!gequal0(rep)) { avma = ltop; return gen_1; } else { avma = ltop; return gen_m1; } avma = ltop; return gen_0; } GEN ideallistfactor(GEN nf, GEN listfact) { pari_sp ltop = avma; GEN Slist = gen_0, S1 = gen_0, test = gen_0, k = gen_0; long l1; /* MODI remove i,j */ if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("entree dans ideallistfactor\n"); Slist = cgetg(1, t_VEC); test = gen_1; l1 = glength(listfact); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i, l2; for (i = 1; i <= l1; ++i) { if (gequal0(gel(listfact, i))) continue; S1 = gcopy(gel(idealfactor(nf, gel(listfact, i)), 1)); l2 = glength(S1); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long j, l3; GEN p4 = gen_0; /* vec */ for (j = 1; j <= l2; ++j) { k = stoi(glength(Slist)); l3 = glength(Slist); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long k; for (k = 1; k <= l3; ++k) { if (gequal(gel(Slist, k), gel(S1, j))) { test = gen_0; break; } if (low_stack(st_lim, stack_lim(btop, 1))) test = gerepilecopy(btop, test); } } if (!gequal0(test)) { p4 = cgetg(2, t_VEC); gel(p4, 1) = gcopy(gel(S1, j)); Slist = concat(Slist, p4); } else test = gen_1; if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 4, &k, &test, &p4, &Slist); } } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 4, &S1, &k, &test, &Slist); } } if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de ideallistfactor\n"); Slist = gerepilecopy(ltop, Slist); return Slist; } long mynfhilbert(GEN nf, GEN a, GEN b, long prec) { pari_sp ltop = avma; GEN al = gen_0, bl = gen_0, S = gen_0; GEN p1 = gen_0; /* vec */ GEN p2 = gen_0; /* int */ GEN p3 = gen_0; /* vec */ long l4; if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) { p1 = cgetg(3, t_VEC); gel(p1, 1) = gcopy(a); gel(p1, 2) = gcopy(b); pari_printf("entree dans mynfhilbert %Ps\n", p1); } if (gequal0(a) || gequal0(b)) pari_err(user, "mynfhilbert : argument = 0"); al = lift(a); bl = lift(b); /* solutions locales aux places reelles */ p2 = icopy(member_r1(nf)); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN i = gen_0; for (i = gen_1; gcmp(i, p2) <= 0; i = gaddgs(i, 1)) { if ((nfsign_s(nf, al, i, prec) < 0) && (nfsign_s(nf, bl, i, prec) < 0)) { if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf("mynfhilbert non soluble a l'infini\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("fin de mynfhilbert\n"); avma = ltop; return -1; } if (low_stack(st_lim, stack_lim(btop, 1))) i = gerepilecopy(btop, i); } } if (typ(a) != t_POLMOD) a = gmodulo(a, member_pol(nf)); if (typ(b) != t_POLMOD) b = gmodulo(b, member_pol(nf)); p3 = cgetg(4, t_VEC); gel(p3, 1) = gen_2; gel(p3, 2) = gcopy(a); gel(p3, 3) = gcopy(b); /* solutions locales aux places finies (celles qui divisent 2ab) */ S = ideallistfactor(nf, p3); l4 = glength(S); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN i = gen_0; long l5 = -1 > 0; /* bool */ for (i = stoi(l4); l5?gcmpgs(i, 2) <= 0:gcmpgs(i, 2) >= 0; i = gaddgs(i, -1)) { /* d'apres la formule du produit on peut eviter un premier */ if (gequalm1(mynfhilbertp(nf, a, b, gel(S, gtos(i)), prec))) { if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf("mynfhilbert non soluble en : %Ps\n", gel(S, gtos(i))); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("fin de mynfhilbert\n"); avma = ltop; return -1; } if (low_stack(st_lim, stack_lim(btop, 1))) i = gerepilecopy(btop, i); } } if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("fin de mynfhilbert\n"); avma = ltop; return 1; } GEN initp(GEN nf, GEN p, long prec) { pari_sp ltop = avma; GEN idval = gen_0, pp = gen_0; GEN p1 = gen_0; /* vec */ if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("entree dans initp\n"); idval = stoi(idealval(nf, gen_2, p)); p1 = cgetg(6, t_VEC); gel(p1, 1) = gcopy(p); gel(p1, 2) = basistoalg(nf, gel(p, 2)); gel(p1, 3) = gcopy(idval); gel(p1, 4) = gen_0; gel(p1, 5) = repres(nf, p, prec); pp = p1; if (!gequal0(idval)) gel(pp, 4) = idealstar0(nf, idealpow0(nf, p, gaddsg(1, gmulsg(2, idval)), 0), 1); else gel(pp, 4) = gdiventgs(gpowgs(member_p(p), pr_get_f(p)), 2); if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de initp\n"); pp = gerepilecopy(ltop, pp); return pp; } GEN deno(GEN num) { pari_sp ltop = avma; GEN p1 = gen_0, p2 = gen_0; if (gequal0(num)) { avma = ltop; return gen_1; } if (typ(num) == t_POL) { p1 = denom(content(num)); p1 = gerepilecopy(ltop, p1); return p1; } p2 = denom(num); p2 = gerepilecopy(ltop, p2); return p2; } GEN nfratpoint(GEN nf, GEN pol, GEN lim, GEN singlepoint, long prec) { pari_sp ltop = avma; GEN compt1 = gen_0, compt2 = gen_0, deg = gen_0, n = gen_0, AA = gen_0, point = gen_0, listpoints = gen_0, vectx = gen_0, evpol = gen_0, sq = gen_0, xpol = gen_0; GEN p1 = gen_0, p2 = gen_0, p3 = gen_0, p4 = gen_0, p5 = gen_0; /* vec */ GEN x = pol_x(fetch_user_var("x")); if (!singlepoint) singlepoint = gen_1; /* MODI remove xx, denoz */ /* MODI add xpol */ if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) { pari_printf("entree dans nfratpoint avec pol = %Ps\n", pol); pari_printf("lim = %Ps\n", lim); } compt1 = gen_0; compt2 = gen_0; deg = stoi(degree(pol)); n = stoi(degree(member_pol(nf))); AA = gshift(lim, 1); if (gequal0(singlepoint)) listpoints = cgetg(1, t_VEC); /* cas triviaux */ sq = nfsqrt(nf, polcoeff0(pol, 0, -1), prec); if (!gequal(sq, cgetg(1, t_VEC))) { p1 = cgetg(4, t_VEC); gel(p1, 1) = gen_0; gel(p1, 2) = gcopy(gel(sq, 1)); gel(p1, 3) = gen_1; point = p1; if (!gequal0(singlepoint)) { if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("fin de nfratpoint\n"); point = gerepilecopy(ltop, point); return point; } p2 = cgetg(2, t_VEC); gel(p2, 1) = gcopy(point); listpoints = concat(listpoints, p2); } sq = nfsqrt(nf, pollead(pol, -1), prec); if (!gequal(sq, cgetg(1, t_VEC))) { p3 = cgetg(4, t_VEC); gel(p3, 1) = gen_1; gel(p3, 2) = gcopy(gel(sq, 1)); gel(p3, 3) = gen_0; point = p3; if (!gequal0(singlepoint)) { if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("fin de nfratpoint\n"); point = gerepilecopy(ltop, point); return point; } p4 = cgetg(2, t_VEC); gel(p4, 1) = gcopy(point); listpoints = concat(listpoints, p4); } /* boucle generale */ point = cgetg(1, t_VEC); { long i; GEN p6 = gen_0; /* vec */ p5 = cgetg(gtos(n)+1, t_VEC); for (i = 1; gcmpsg(i, n) <= 0; ++i) { p6 = cgetg(3, t_VEC); gel(p6, 1) = gneg(lim); gel(p6, 2) = gcopy(lim); gel(p5, i) = p6; } } vectx = p5; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN denoz = gen_0; for (denoz = gen_1; gcmp(denoz, lim) <= 0; denoz = gaddgs(denoz, 1)) { { GEN fv_data = gen_0; GEN (*fv_next)(GEN, GEN); /* func_GG */ GEN xx = forvec_start(vectx, 0, &fv_data, &fv_next); /* vec */ { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN p7 = gen_0, p8 = gen_0; /* vec */ for ( ; xx; xx = fv_next(fv_data, xx)) { if (gequal1(denoz) || gequal1(ggcd(content(xx), denoz))) { xpol = basistoalg(nf, gtrans(xx)); evpol = gsubst(pol, gvar(x), gdiv(xpol, denoz)); sq = nfsqrt(nf, evpol, prec); if (!gequal(sq, cgetg(1, t_VEC))) { p7 = cgetg(4, t_VEC); gel(p7, 1) = gdiv(xpol, denoz); gel(p7, 2) = gcopy(gel(sq, 1)); gel(p7, 3) = gen_1; point = p7; if (!gequal0(singlepoint)) goto label5; p8 = cgetg(2, t_VEC); gel(p8, 1) = gcopy(point); listpoints = concat(listpoints, p8); } } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 8, &xpol, &evpol, &sq, &p7, &point, &p8, &listpoints, &xx); } } } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 6, &denoz, &xpol, &evpol, &sq, &point, &listpoints); } label5:; } if (!gequal0(singlepoint)) listpoints = point; if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("sortie de nfratpoint\n"); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf("points trouves par nfratpoint = %Ps\n", listpoints); listpoints = gerepilecopy(ltop, listpoints); return listpoints; } GEN repres(GEN nf, GEN p, long prec) { pari_sp ltop = avma; GEN fond = gen_0, mat = gen_0, f = gen_0, rep = gen_0, pp = gen_0, ppi = gen_0, pp2 = gen_0, jppi = gen_0, gjf = gen_0; long l1; GEN p2 = gen_0; GEN p3 = gen_0; /* vec */ GEN p4 = gen_0; /* MODI remove i,j,k */ if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("entree dans repres\n"); fond = cgetg(1, t_VEC); mat = idealhnf0(nf, p, NULL); l1 = glength(mat); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; for (i = 1; i <= l1; ++i) { if (!gequalgs(gcoeff(mat, i, i), 1)) fond = concat(fond, gel(member_zk(nf), i)); if (low_stack(st_lim, stack_lim(btop, 1))) fond = gerepilecopy(btop, fond); } } f = stoi(glength(fond)); pp = icopy(member_p(p)); p2 = gpow(pp, f, prec); { long i; p3 = cgetg(gtos(p2)+1, t_VEC); for (i = 1; gcmpsg(i, p2) <= 0; ++i) gel(p3, i) = gen_0; } rep = p3; gel(rep, 1) = gen_0; ppi = gen_1; pp2 = gdiventgs(pp, 2); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN i = gen_0, p5 = gen_0; for (i = gen_1; gcmp(i, f) <= 0; i = gaddgs(i, 1)) { p5 = gsubgs(pp, 1); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN j = gen_0, p6 = gen_0; for (j = gen_1; gcmp(j, p5) <= 0; j = gaddgs(j, 1)) { if (gcmp(j, pp2) <= 0) gjf = gmul(j, gel(fond, gtos(i))); else gjf = gmul(gsub(j, pp), gel(fond, gtos(i))); jppi = gmul(j, ppi); p6 = gsubgs(ppi, 1); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN k = gen_0; for (k = gen_0; gcmp(k, p6) <= 0; k = gaddgs(k, 1)) { gel(rep, gtos(gaddgs(gadd(jppi, k), 1))) = gadd(gel(rep, gtos(gaddgs(k, 1))), gjf); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &k, &rep); } } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 5, &j, &gjf, &jppi, &p6, &rep); } } ppi = gmul(ppi, pp); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 6, &i, &p5, &gjf, &jppi, &rep, &ppi); } } if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de repres\n"); p4 = gmodulo(rep, member_pol(nf)); p4 = gerepilecopy(ltop, p4); return p4; } GEN val(GEN nf, GEN num, GEN p) { pari_sp ltop = avma; GEN p1 = gen_0; if (gequal0(num)) p1 = BIGINT; else p1 = stoi(idealval(nf, lift(num), p)); p1 = gerepilecopy(ltop, p1); return p1; } GEN nfissquarep(GEN nf, GEN a, GEN p, GEN q, long prec) { pari_sp ltop = avma; GEN pherm = gen_0, f = gen_0, aaa = gen_0, n = gen_0, pp = gen_0, qq = gen_0, e = gen_0, z = gen_0, xx = gen_0, yy = gen_0, r = gen_0, aux = gen_0, b = gen_0, m = gen_0, vp = gen_0, inv2x = gen_0, zinit = gen_0, zlog = gen_0, expo = gen_0, id = gen_0; long l1; /* MODI add id */ if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("entree dans nfissquarep %Ps%Ps%Ps\n", a, p, q); if (gequal0(a) || gequal1(a)) { if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("fin de nfissquarep\n"); a = gerepilecopy(ltop, a); return a; } pherm = idealhnf0(nf, p, NULL); if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("pherm = %Ps\n", pherm); f = stoi(idealval(nf, a, p)); if (gcmp(f, q) >= 0) { if (gcmp(f, q) > 0) aaa = gpow(basistoalg(nf, gel(p, 2)), gshift(gaddgs(q, 1), -1), prec); else aaa = gen_0; if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("fin de nfissquarep\n"); aaa = gerepilecopy(ltop, aaa); return aaa; } if (!gequal0(f)) aaa = gmul(a, gpow(basistoalg(nf, gdiv(gel(p, 5), member_p(p))), f, prec)); else aaa = gcopy(a); if (!gequalgs(gcoeff(pherm, 1, 1), 2)) { /* cas ou p ne divise pas 2 */ /* algorithme de Shanks */ n = nfrandintmodid(nf, pherm); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); while (nfpsquareodd(nf, n, p, prec)) { n = nfrandintmodid(nf, pherm); if (low_stack(st_lim, stack_lim(btop, 1))) n = gerepilecopy(btop, n); } } pp = gmodulsg(1, member_p(p)); n = gmul(n, pp); qq = gdiventgs(idealnorm(nf, pherm), 2); e = gen_1; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); while (gequal0(gmodgs(qq, 2))) { e = gaddgs(e, 1); qq = gdivent(qq, gen_2); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &e, &qq); } } z = mynfeltreduce(nf, lift(lift(gpow(n, qq, prec))), pherm); yy = z; r = e; xx = mynfeltreduce(nf, lift(lift(gpow(gmul(aaa, pp), gdiventgs(qq, 2), prec))), pherm); aux = mynfeltreduce(nf, gmul(aaa, xx), pherm); b = mynfeltreduce(nf, gmul(aux, xx), pherm); xx = aux; aux = b; m = gen_0; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); while (gequal0(val(nf, gsubgs(aux, 1), p))) { m = gaddgs(m, 1); aux = mynfeltreduce(nf, gsqr(aux), pherm); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &m, &aux); } } { pari_sp btop = avma, st_lim = stack_lim(btop, 1); while (!gequal0(m)) { if (gequal(m, r)) pari_err(user, "nfissquarep : m = r"); yy = gmul(yy, pp); aux = mynfeltreduce(nf, lift(lift(powgi(yy, shifti(gen_1, gtos(gsubgs(gsub(r, m), 1)))))), pherm); yy = mynfeltreduce(nf, gsqr(aux), pherm); r = m; xx = mynfeltreduce(nf, gmul(xx, aux), pherm); b = mynfeltreduce(nf, gmul(b, yy), pherm); aux = b; m = gen_0; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); while (gequal0(val(nf, gsubgs(aux, 1), p))) { m = gaddgs(m, 1); aux = mynfeltreduce(nf, gsqr(aux), pherm); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &m, &aux); } } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 6, &yy, &aux, &r, &xx, &b, &m); } } /* lift de Hensel */ /* */ if (gcmpgs(q, 1) > 0) { vp = stoi(idealval(nf, gsub(gsqr(xx), aaa), p)); if (gcmp(vp, gsub(q, f)) < 0) { yy = gmulsg(2, xx); inv2x = gdiv(basistoalg(nf, gel(idealaddtoone0(nf, yy, p), 1)), yy); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); while (gcmp(vp, q) < 0) { vp = gaddgs(vp, 1); xx = gsub(xx, gmul(gsub(gsqr(xx), aaa), inv2x)); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &vp, &xx); } } } if (!gequal0(f)) xx = gmul(xx, gpow(basistoalg(nf, gel(p, 2)), gshift(f, -1), prec)); } xx = mynfeltreduce(nf, xx, idealpow0(nf, p, q, 0)); } else { /* cas ou p divise 2 *\ */ if (gcmpgs(gsub(q, f), 1) > 0) id = idealpow0(nf, p, gsub(q, f), 0); else id = pherm; zinit = idealstar0(nf, id, 2); zlog = ideallog(nf, aaa, zinit); xx = gen_1; l1 = glength(zlog); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; for (i = 1; i <= l1; ++i) { expo = gcopy(gel(zlog, i)); if (!gequal0(expo)) { if (smodss(gequal0(expo), 2)) expo = gshift(expo, -1); else { aux = gcopy(gel(gel(zinit, 2), i)); expo = gmod(gmul(expo, gshift(gaddgs(aux, 1), -1)), aux); } xx = gmul(xx, gpow(basistoalg(nf, gel(gel(gel(zinit, 2), 3), i)), expo, prec)); } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 3, &expo, &aux, &xx); } } if (!gequal0(f)) { xx = gmul(xx, gpow(basistoalg(nf, gel(p, 2)), gshift(f, -1), prec)); id = idealpow0(nf, p, q, 0); } xx = mynfeltreduce(nf, xx, id); } if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("fin de nfissquarep %Ps\n", xx); xx = gerepilecopy(ltop, xx); return xx; } long nfpsquareodd(GEN nf, GEN a, GEN p, long prec) { pari_sp ltop = avma; GEN v = gen_0, ap = gen_0, norme = gen_0, den = gen_0; if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("entree dans nfpsquareodd\n"); if (gequal0(a)) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nfpsquareodd\n"); avma = ltop; return 1; } v = stoi(idealval(nf, lift(a), p)); if (!gequal0(gmodgs(v, 2))) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nfpsquareodd\n"); avma = ltop; return 0; } ap = gdiv(a, gpow(basistoalg(nf, gel(p, 2)), v, prec)); norme = gdiventgs(idealnorm(nf, p), 2); den = gmod(denom(content(lift(ap))), member_p(p)); if (gsigne(den)) ap = gmul(ap, gmodulsg(1, member_p(p))); ap = gsubgs(gpow(ap, norme, prec), 1); if (gequal0(ap)) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nfpsquareodd\n"); avma = ltop; return 1; } ap = lift(lift(ap)); if (idealval(nf, ap, p) > 0) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nfpsquareodd\n"); avma = ltop; return 1; } if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nfpsquareodd\n"); avma = ltop; return 0; } long nfpsquare(GEN nf, GEN a, GEN p, GEN zinit, long prec) { pari_sp ltop = avma; GEN valap = gen_0, zlog = gen_0; GEN p1 = gen_0; /* vec */ long l2, l3; /* MODI remove i */ if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) { p1 = cgetg(4, t_VEC); gel(p1, 1) = gcopy(a); gel(p1, 2) = gcopy(p); gel(p1, 3) = gcopy(zinit); pari_printf("entree dans nfpsquare %Ps\n", p1); } if (gequal0(a)) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nfpsquare\n"); avma = ltop; return 1; } if (cmpis(member_p(p), 2) != 0) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nfpsquare\n"); l2 = nfpsquareodd(nf, a, p, prec); avma = ltop; return l2; } valap = stoi(idealval(nf, a, p)); if (!gequal0(gmodgs(valap, 2))) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nfpsquare\n"); avma = ltop; return 0; } if (!gequal0(valap)) zlog = ideallog(nf, gmul(a, gpow(gdiv(basistoalg(nf, gel(p, 5)), member_p(p)), valap, prec)), zinit); else zlog = ideallog(nf, a, zinit); l3 = glength(gel(gel(zinit, 2), 2)); { pari_sp btop = avma; long i; for (i = 1; i <= l3; ++i) { if ((gequal0(gmodgs(gel(gel(gel(zinit, 2), 2), i), 2))) && !gequal0(gmodgs(gel(zlog, i), 2))) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nfpsquare\n"); avma = ltop; return 0; } avma = btop; } } if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nfpsquare\n"); avma = ltop; return 1; } long nfpsquareq(GEN nf, GEN a, GEN p, GEN q, long prec) { pari_sp ltop = avma; GEN vala = gen_0, zinit = gen_0, zlog = gen_0; GEN p1 = gen_0; /* vec */ long l2; /* MODI remove i */ if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) { p1 = cgetg(4, t_VEC); gel(p1, 1) = gcopy(a); gel(p1, 2) = gcopy(p); gel(p1, 3) = gcopy(q); pari_printf("entree dans nfpsquareq %Ps\n", p1); } if (gequal0(a)) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nfpsquareq\n"); avma = ltop; return 1; } vala = stoi(idealval(nf, a, p)); if (gcmp(vala, q) >= 0) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nfpsquareq\n"); avma = ltop; return 1; } if (!gequal0(gmodgs(vala, 2))) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nfpsquareq\n"); avma = ltop; return 0; } zinit = idealstar0(nf, idealpow0(nf, p, gsub(q, vala), 0), 2); zlog = ideallog(nf, gmul(a, gpow(basistoalg(nf, gdivgs(gel(p, 5), 2)), vala, prec)), zinit); l2 = glength(gel(gel(zinit, 2), 2)); { pari_sp btop = avma; long i; for (i = 1; i <= l2; ++i) { if ((gequal0(gmodgs(gel(gel(gel(zinit, 2), 2), i), 2))) && !gequal0(gmodgs(gel(zlog, i), 2))) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nfpsquareq\n"); avma = ltop; return 0; } avma = btop; } } if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nfpsquareq\n"); avma = ltop; return 1; } long nflemma6(GEN nf, GEN pol, GEN p, GEN nu, GEN xx, long prec) { pari_sp ltop = avma; GEN gx = gen_0, gpx = gen_0, lambda = gen_0, mu = gen_0, x = pol_x(fetch_user_var("x")); if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("entree dans nflemma6\n"); gx = gsubst(pol, gvar(x), xx); if (nfpsquareodd(nf, gx, p, prec)) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nflemma6\n"); avma = ltop; return 1; } gpx = gsubst(deriv(pol,-1), gvar(x), xx); lambda = val(nf, gx, p); mu = val(nf, gpx, p); if (gcmp(lambda, gmulsg(2, mu)) > 0) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nflemma6\n"); avma = ltop; return 1; } if ((gcmp(lambda, gmulsg(2, nu)) >= 0) && (gcmp(mu, nu) >= 0)) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nflemma6\n"); avma = ltop; return 0; } if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nflemma6\n"); avma = ltop; return -1; } long nflemma7(GEN nf, GEN pol, GEN p, GEN nu, GEN xx, GEN zinit, long prec) { pari_sp ltop = avma; GEN gx = gen_0, gpx = gen_0, v = gen_0, lambda = gen_0, mu = gen_0, q = gen_0; GEN p1 = gen_0; /* vec */ GEN x = pol_x(fetch_user_var("x")); if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) { p1 = cgetg(3, t_VEC); gel(p1, 1) = gcopy(xx); gel(p1, 2) = gcopy(nu); pari_printf("entree dans nflemma7 %Ps\n", p1); } gx = gsubst(pol, gvar(x), xx); if (nfpsquare(nf, gx, p, zinit, prec)) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nflemma7\n"); avma = ltop; return 1; } gpx = gsubst(deriv(pol,-1), gvar(x), xx); v = gcopy(gel(p, 3)); lambda = val(nf, gx, p); mu = val(nf, gpx, p); if (gcmp(lambda, gmulsg(2, mu)) > 0) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nflemma7\n"); avma = ltop; return 1; } if (gcmp(nu, mu) > 0) { if (!gequal0(gmodgs(lambda, 2))) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nflemma7\n"); avma = ltop; return -1; } q = gsub(gadd(mu, nu), lambda); if (gcmp(q, gmulsg(2, v)) > 0) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nflemma7\n"); avma = ltop; return -1; } if (nfpsquareq(nf, gmul(gx, gpow(basistoalg(nf, gdivgs(gel(p, 5), 2)), lambda, prec)), p, q, prec)) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nflemma7\n"); avma = ltop; return 1; } } else { if (gcmp(lambda, gmulsg(2, nu)) >= 0) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nflemma7\n"); avma = ltop; return 0; } if (!gequal0(gmodgs(lambda, 2))) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nflemma7\n"); avma = ltop; return -1; } q = gsub(gmulsg(2, nu), lambda); if (gcmp(q, gmulsg(2, v)) > 0) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nflemma7\n"); avma = ltop; return -1; } if (nfpsquareq(nf, gmul(gx, gpow(basistoalg(nf, gdivgs(gel(p, 5), 2)), lambda, prec)), p, q, prec)) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nflemma7\n"); avma = ltop; return 0; } } if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nflemma7\n"); avma = ltop; return -1; } long nfzpsoluble(GEN nf, GEN pol, GEN p, GEN nu, GEN pnu, GEN x0, long prec) { pari_sp ltop = avma; GEN result = gen_0, pnup = gen_0, lrep = gen_0; GEN p1 = gen_0; /* vec */ if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) { p1 = cgetg(3, t_VEC); gel(p1, 1) = lift(x0); gel(p1, 2) = gcopy(nu); pari_printf("entree dans nfzpsoluble %Ps\n", p1); } if (gequal0(gel(p, 3))) result = stoi(nflemma6(nf, pol, gel(p, 1), nu, x0, prec)); else result = stoi(nflemma7(nf, pol, gel(p, 1), nu, x0, gel(p, 4), prec)); if (gequalgs(result, 1)) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nfzpsoluble\n"); avma = ltop; return 1; } if (gequalm1(result)) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nfzpsoluble\n"); avma = ltop; return 0; } pnup = gmul(pnu, gel(p, 2)); lrep = stoi(glength(gel(p, 5))); nu = gaddgs(nu, 1); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN i = gen_0; for (i = gen_1; gcmp(i, lrep) <= 0; i = gaddgs(i, 1)) { if (nfzpsoluble(nf, pol, p, nu, pnup, gadd(x0, gmul(pnu, gel(gel(p, 5), gtos(i)))), prec)) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nfzpsoluble\n"); avma = ltop; return 1; } if (low_stack(st_lim, stack_lim(btop, 1))) i = gerepilecopy(btop, i); } } if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nfzpsoluble\n"); avma = ltop; return 0; } GEN mynfeltmod(GEN nf, GEN a, GEN b) { pari_sp ltop = avma; GEN qred = gen_0; qred = ground(algtobasis(nf, gdiv(a, b))); qred = gsub(a, gmul(b, basistoalg(nf, qred))); qred = gerepilecopy(ltop, qred); return qred; } GEN mynfeltreduce(GEN nf, GEN a, GEN id) { pari_sp ltop = avma; GEN p1 = gen_0; p1 = basistoalg(nf, nfreduce(nf, algtobasis(nf, a), id)); p1 = gerepilecopy(ltop, p1); return p1; } GEN nfrandintmodid(GEN nf, GEN id) { pari_sp ltop = avma; GEN res = gen_0; if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("entree dans nfrandintmodid\n"); res = gen_0; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); while (gequal0(res)) { res = nfrandint(nf, gen_0); res = mynfeltreduce(nf, res, id); if (low_stack(st_lim, stack_lim(btop, 1))) res = gerepilecopy(btop, res); } } if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nfrandintmodid\n"); res = gerepilecopy(ltop, res); return res; } GEN nfrandint(GEN nf, GEN borne) { pari_sp ltop = avma; GEN l = gen_0, res = gen_0; GEN p1 = gen_0; /* vec */ /* MODI remove i */ if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("entree dans nfrandint\n"); l = stoi(lg(member_zk(nf))-1); { long i; p1 = cgetg(gtos(l)+1, t_COL); for (i = 1; gcmpsg(i, l) <= 0; ++i) gel(p1, i) = gen_0; } res = p1; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN i = gen_0; for (i = gen_1; gcmp(i, l) <= 0; i = gaddgs(i, 1)) { if (!gequal0(borne)) gel(res, gtos(i)) = gsub(genrand(gshift(borne, 1)), borne); else gel(res, gtos(i)) = genrand(NULL); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &i, &res); } } res = basistoalg(nf, res); if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nfrandint\n"); res = gerepilecopy(ltop, res); return res; } long nfqpsolublebig(GEN nf, GEN pol, GEN p, GEN ap, GEN b, long prec) { pari_sp ltop = avma; GEN deg = gen_0, xx = gen_0, z = gen_0, Px = gen_0, cont = gen_0, pi = gen_0, pol2 = gen_0, Roots = gen_0; long l1; GEN x = pol_x(fetch_user_var("x")); if (!ap) ap = gen_0; if (!b) b = gen_1; /* MODI remove i,j */ if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("entree dans nfqpsolublebig avec %Ps\n", member_p(p)); deg = stoi(degree(pol)); if (nfpsquareodd(nf, polcoeff0(pol, 0, -1), p, prec)) { if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("fin de nfqpsolublebig\n"); avma = ltop; return 1; } if (nfpsquareodd(nf, pollead(pol, -1), p, prec)) { if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("fin de nfqpsolublebig\n"); avma = ltop; return 1; } /* on tient compte du contenu de pol */ cont = stoi(idealval(nf, polcoeff0(pol, 0, -1), p)); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN i = gen_0; for (i = gen_1; gcmp(i, deg) <= 0; i = gaddgs(i, 1)) { if (!gequal0(cont)) cont = gmings(cont, idealval(nf, polcoeff0(pol, gtos(i), -1), p)); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &i, &cont); } } if (!gequal0(cont)) pi = basistoalg(nf, gdiv(gel(p, 5), member_p(p))); if (gcmpgs(cont, 1) > 0) pol = gmul(pol, gpow(pi, gmulsg(2, gdiventgs(cont, 2)), prec)); /* On essaye des valeurs de x au hasard */ if (!gequal0(gmodgs(cont, 2))) pol2 = gmul(pol, pi); else { pol2 = gcopy(pol); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN i = gen_0, p2 = gen_0; for (i = gen_1; gcmp(i, MAXPROB) <= 0; i = gaddgs(i, 1)) { xx = nfrandint(nf, gen_0); z = gen_0; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); while (gequal0(z)) { z = genrand(NULL); if (low_stack(st_lim, stack_lim(btop, 1))) z = gerepilecopy(btop, z); } } xx = gadd(gmul(gneg(ap), z), gmul(b, xx)); Px = polcoeff0(pol, gtos(deg), -1); p2 = gsubgs(deg, 1); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN j = gen_0; long l3 = -1 > 0; /* bool */ for (j = p2; l3?gcmpgs(j, 0) <= 0:gcmpgs(j, 0) >= 0; j = gaddgs(j, -1)) { Px = gadd(gmul(Px, xx), polcoeff0(pol, gtos(j), -1)); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &j, &Px); } } Px = gmul(Px, gpow(z, deg, prec)); if (nfpsquareodd(nf, Px, p, prec)) { if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("fin de nfqpsolublebig\n"); avma = ltop; return 1; } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 5, &i, &xx, &z, &Px, &p2); } } } /* On essaye les racines de pol */ Roots = nfpolrootsmod(nf, pol2, p); pi = basistoalg(nf, gel(p, 2)); l1 = glength(Roots); { pari_sp btop = avma; long i; for (i = 1; i <= l1; ++i) { if (nfqpsolublebig(nf, gsubst(pol, gvar(x), gadd(gmul(pi, x), gel(Roots, i))), p, NULL, NULL, prec)) { avma = ltop; return 1; } avma = btop; } } if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("fin de nfqpsolublebig\n"); avma = ltop; return 0; } GEN nfpolrootsmod(GEN nf, GEN pol, GEN p) { pari_sp ltop = avma; GEN factlist = gen_0, sol = gen_0; long l1; factlist = gcopy(gel(nffactormod(nf, pol, p), 1)); sol = cgetg(1, t_VEC); l1 = glength(factlist); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; GEN p2 = gen_0; /* vec */ for (i = 1; i <= l1; ++i) { if (degree(gel(factlist, i)) == 1) { p2 = cgetg(2, t_VEC); gel(p2, 1) = gdiv(gneg(polcoeff0(gel(factlist, i), 0, -1)), polcoeff0(gel(factlist, i), 1, -1)); sol = concat(sol, p2); } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &p2, &sol); } } sol = gerepilecopy(ltop, sol); return sol; } long nfqpsoluble(GEN nf, GEN pol, GEN p, long prec) { pari_sp ltop = avma; if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("entree dans nfqpsoluble %Ps\n", p); if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("pol = %Ps\n", pol); if (nfpsquare(nf, pollead(pol, -1), gel(p, 1), gel(p, 4), prec)) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nfqpsoluble\n"); avma = ltop; return 1; } if (nfpsquare(nf, polcoeff0(pol, 0, -1), gel(p, 1), gel(p, 4), prec)) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nfqpsoluble\n"); avma = ltop; return 1; } if (nfzpsoluble(nf, pol, p, gen_0, gen_1, gen_0, prec)) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nfqpsoluble\n"); avma = ltop; return 1; } if (nfzpsoluble(nf, polrecip(pol), p, gen_1, gel(p, 2), gen_0, prec)) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nfqpsoluble\n"); avma = ltop; return 1; } if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de nfqpsoluble\n"); avma = ltop; return 0; } long nflocallysoluble(GEN nf, GEN pol, GEN r, GEN a, GEN b, long prec) { pari_sp ltop = avma; GEN pol0 = gen_0, plist = gen_0, add = gen_0, ff = gen_0, p = gen_0, Delta = gen_0, vecpol = gen_0, vecpolr = gen_0, Sturmr = gen_0; GEN p1 = gen_0; /* vec */ GEN p2 = gen_0; /* int */ if (!r) r = gen_0; if (!a) a = gen_1; if (!b) b = gen_1; if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) { p1 = cgetg(5, t_VEC); gel(p1, 1) = gcopy(pol); gel(p1, 2) = gcopy(r); gel(p1, 3) = gcopy(a); gel(p1, 4) = gcopy(b); pari_printf("entree dans nflocallysoluble %Ps\n", p1); } pol0 = gcopy(pol); /* */ /* places finies de plist *\ */ /* */ pol = gmul(/* */ /* places finies de plist *\ */ /* */ pol, gsqr(deno(content(lift(pol))))); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long ii, l3; for (ii = 1; ii <= 3; ++ii) { if (ii == 1) plist = idealprimedec(nf, gen_2); if ((ii == 2) && !gequal0(r)) plist = gcopy(gel(idealfactor(nf, gdiv(gdiv(poldisc0(gdiv(pol0, pollead(pol0, -1)), -1), gpowgs(pollead(pol0, -1), 6)), gpowgs(gen_2, 12))), 1)); if ((ii == 2) && (gequal0(r))) plist = gcopy(gel(idealfactor(nf, poldisc0(pol0, -1)), 1)); if (ii == 3) { add = idealadd(nf, a, b); ff = gcopy(gel(factor(idealnorm(nf, add)), 1)); addprimes(ff); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("liste de premiers = %Ps\n", ff); plist = gcopy(gel(idealfactor(nf, add), 1)); } l3 = glength(plist); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; for (i = 1; i <= l3; ++i) { p = gcopy(gel(plist, i)); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf("p = %Ps\n", p); if (gcmp(member_p(p), LIMBIGPRIME) < 0) { if (!nfqpsoluble(nf, pol, initp(nf, p, prec), prec)) { if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" non ELS en %Ps\n", p); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("fin de nflocallysoluble\n"); avma = ltop; return 0; } } else if (!nfqpsolublebig(nf, pol, p, gdiv(r, a), b, prec)) { if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" non ELS en %Ps ( = grand premier )\n", member_p(p)); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("fin de nflocallysoluble\n"); avma = ltop; return 0; } if (low_stack(st_lim, stack_lim(btop, 1))) p = gerepilecopy(btop, p); } } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 4, &plist, &add, &ff, &p); } } /* places reelles */ if (signe(member_r1(nf))) { Delta = poldisc0(pol, -1); vecpol = gtovec(pol); p2 = icopy(member_r1(nf)); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN i = gen_0; long l4; GEN p5 = gen_0; /* vec */ for (i = gen_1; gcmp(i, p2) <= 0; i = gaddgs(i, 1)) { if (nfsign_s(nf, pollead(pol, -1), i, prec) > 0) continue; if (nfsign_s(nf, polcoeff0(pol, 0, -1), i, prec) > 0) continue; if (nfsign_s(nf, Delta, i, prec) < 0) continue; l4 = glength(vecpol); { long j; p5 = cgetg(l4+1, t_VEC); for (j = 1; j <= l4; ++j) gel(p5, j) = mysubst(gel(vecpol, j), gel(member_roots(nf), gtos(i))); } vecpolr = p5; Sturmr = stoi(sturmpart(gtopoly(vecpolr, -1), NULL, NULL)); if (gequal0(Sturmr)) { if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" non ELS a l'infini\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("fin de nflocallysoluble\n"); avma = ltop; return 0; } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 4, &i, &p5, &vecpolr, &Sturmr); } } } if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" quartique ELS \n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("fin de nflocallysoluble\n"); avma = ltop; return 1; } GEN nfellcount(GEN nf, GEN c, GEN d, GEN KS2gen, GEN pointstriv, long prec) /* vec */ { pari_sp ltop = avma; GEN found = gen_0, listgen = gen_0, listpointscount = gen_0, m1 = gen_0, m2 = gen_0, lastloc = gen_0, mask = gen_0, i = gen_0, d1 = gen_0, iaux = gen_0, j = gen_0, triv = gen_0, pol = gen_0, point = gen_0, deuxpoints = gen_0, aux = gen_0, v = gen_0; GEN p1 = gen_0; /* vec */ long l2; GEN x = pol_x(fetch_user_var("x")); GEN p3 = gen_0, p4 = gen_0; /* vec */ /* MODI add aux, v */ if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) { p1 = cgetg(3, t_VEC); gel(p1, 1) = gcopy(c); gel(p1, 2) = gcopy(d); pari_printf("entree dans nfellcount %Ps\n", p1); } found = gen_0; listgen = gcopy(KS2gen); listpointscount = cgetg(1, t_VEC); m1 = m2 = gen_0; lastloc = gen_m1; mask = shifti(gen_1, glength(KS2gen)); i = gen_1; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long l5; GEN p6 = gen_0, p7 = gen_0, p8 = gen_0, p9 = gen_0, p10 = gen_0, p11 = gen_0; /* vec */ while (gcmp(i, mask) < 0) { d1 = gen_1; iaux = i; j = gen_1; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); while (!gequal0(iaux)) { if (!gequal0(gmodgs(iaux, 2))) d1 = gmul(d1, gel(listgen, gtos(j))); iaux = gshift(iaux, -1); j = gaddgs(j, 1); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 3, &d1, &iaux, &j); } } if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf("d1 = %Ps\n", d1); triv = gen_0; l5 = glength(pointstriv); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long j; GEN p12 = gen_0; /* vec */ for (j = 1; j <= l5; ++j) { if (!gequal0(gmul(gel(gel(pointstriv, j), 3), gel(gel(pointstriv, j), 1))) && nfissquare(nf, gmul(gmul(d1, gel(gel(pointstriv, j), 1)), gel(gel(pointstriv, j), 3)), prec)) { p12 = cgetg(2, t_VEC); gel(p12, 1) = gcopy(gel(pointstriv, j)); listpointscount = concat(listpointscount, p12); if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf("point trivial\n"); triv = gen_1; m1 = gaddgs(m1, 1); if (gcmp(degre(i), lastloc) > 0) m2 = gaddgs(m2, 1); found = gen_1; lastloc = gen_m1; break; } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 7, &p12, &listpointscount, &triv, &m1, &m2, &found, &lastloc); } } if (gequal0(triv)) { p6 = cgetg(6, t_VEC); gel(p6, 1) = gcopy(d1); gel(p6, 2) = gen_0; gel(p6, 3) = gcopy(c); gel(p6, 4) = gen_0; gel(p6, 5) = gdiv(d, d1); pol = gtopoly(p6, -1); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf("quartique = y^2 = %Ps\n", pol); point = nfratpoint(nf, pol, LIM1, gen_1, prec); if (!gequal(point, cgetg(1, t_VEC))) { if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf("point sur la quartique\n"); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf("%Ps\n", point); m1 = gaddgs(m1, 1); if (!gequalgs(gel(point, 3), 0)) { aux = gdiv(gmul(d1, gel(point, 1)), gsqr(gel(point, 3))); p7 = cgetg(3, t_VEC); gel(p7, 1) = gmul(aux, gel(point, 1)); gel(p7, 2) = gdiv(gmul(aux, gel(point, 2)), gel(point, 3)); deuxpoints = p7; } else { p8 = cgetg(2, t_VEC); gel(p8, 1) = gen_0; deuxpoints = p8; } p9 = cgetg(2, t_VEC); gel(p9, 1) = gcopy(deuxpoints); listpointscount = concat(listpointscount, p9); if (gcmp(degre(i), lastloc) > 0) m2 = gaddgs(m2, 1); found = gen_1; lastloc = gen_m1; } else if (nflocallysoluble(nf, pol, NULL, NULL, NULL, prec)) { if (gcmp(degre(i), lastloc) > 0) { m2 = gaddgs(m2, 1); lastloc = degre(i); } point = nfratpoint(nf, pol, LIM3, gen_1, prec); if (!gequal(point, cgetg(1, t_VEC))) { if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf("point sur la quartique\n"); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf("%Ps\n", point); m1 = gaddgs(m1, 1); aux = gdiv(gmul(d1, gel(point, 1)), gsqr(gel(point, 3))); p10 = cgetg(3, t_VEC); gel(p10, 1) = gmul(aux, gel(point, 1)); gel(p10, 2) = gdiv(gmul(aux, gel(point, 2)), gel(point, 3)); deuxpoints = p10; p11 = cgetg(2, t_VEC); gel(p11, 1) = gcopy(deuxpoints); listpointscount = concat(listpointscount, p11); if (gcmp(degre(i), lastloc) > 0) m2 = gaddgs(m2, 1); found = gen_1; lastloc = gen_m1; } else if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf("pas de point trouve sur la quartique\n"); } } if (!gequal0(found)) { found = gen_0; v = gen_0; iaux = gshift(i, -1); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); while (!gequal0(iaux)) { iaux = gshift(iaux, -1); v = gaddgs(v, 1); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &iaux, &v); } } mask = gshift(mask, -1); listgen = extract0(listgen, subis(subii(shifti(gen_1, glength(listgen)), shifti(gen_1, gtos(v))), 1), NULL); i = shifti(gen_1, gtos(v)); } else i = gaddgs(i, 1); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 23, &d1, &iaux, &j, &triv, &listpointscount, &m1, &m2, &found, &lastloc, &p6, &pol, &point, &aux, &p7, &deuxpoints, &p8, &p9, &p10, &p11, &v, &mask, &listgen, &i); } } l2 = glength(listpointscount); { pari_sp btop = avma; long i; for (i = 1; i <= l2; ++i) { if (glength(gel(listpointscount, i)) > 1) if (!gequalgs(gsub(gsubst(gadd(gadd(gpowgs(x, 3), gmul(c, gsqr(x))), gmul(d, x)), gvar(x), gel(gel(listpointscount, i), 1)), gsqr(gel(gel(listpointscount, i), 2))), 0)) pari_err(user, "nfellcount : MAUVAIS POINT = %Ps", gel(listpointscount, i)); avma = btop; } } if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("fin de nfellcount\n"); p3 = cgetg(3, t_VEC); gel(p3, 1) = gcopy(listpointscount); p4 = cgetg(3, t_VEC); gel(p4, 1) = gcopy(m1); gel(p4, 2) = gcopy(m2); gel(p3, 2) = p4; p3 = gerepilecopy(ltop, p3); return p3; } /* To keep gp2c happy */ GEN gettufu(GEN bnf) { pari_sp ltop = avma; GEN p1 = gen_0; p1 = concat(gel(member_tu(bnf), 2), member_fu(bnf)); p1 = gerepilecopy(ltop, p1); return p1; } GEN getfutu(GEN bnf) { pari_sp ltop = avma; GEN p1 = gen_0; p1 = concat(member_fu(bnf), gel(member_tu(bnf), 2)); p1 = gerepilecopy(ltop, p1); return p1; } GEN bnfell2descent_viaisog(GEN bnf, GEN ell, long prec) /* vec */ { pari_sp ltop = avma; GEN P = gen_0, Pfact = gen_0, tors = gen_0, pointstriv = gen_0, apinit = gen_0, bpinit = gen_0, plist = gen_0, KS2prod = gen_0, oddclass = gen_0, KS2gen = gen_0, listpoints = gen_0, pointgen = gen_0, n1 = gen_0, n2 = gen_0, certain = gen_0, np1 = gen_0, np2 = gen_0, listpoints2 = gen_0, aux1 = gen_0, aux2 = gen_0, certainp = gen_0, rang = gen_0, strange = gen_0, y = pol_x(fetch_user_var("y")); GEN p1 = gen_0, p2 = gen_0, p3 = gen_0, p4 = gen_0, p5 = gen_0, p6 = gen_0, p7 = gen_0; /* vec */ GEN x = pol_x(fetch_user_var("x")); long l8, l9; GEN p10 = gen_0, p11 = gen_0, p12 = gen_0, p13 = gen_0, p14 = gen_0, p15 = gen_0, p16 = gen_0; /* vec */ long l17; GEN p18 = gen_0; /* vec */ long l19; GEN p20 = gen_0; /* vec */ /* MODI remove i */ if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf("Algorithme de la 2-descente par isogenies\n"); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf("entree dans bnfell2descent_viaisog\n"); if (!gequal(gpolvar(member_pol(bnf)), y)) pari_err(user, " bnfell2descent_viaisog : la variable du corps de nombres doit etre y "); ell = smallellinit(gmodulo(lift(ell), member_pol(bnf))); if (gequal0(member_disc(ell))) pari_err(user, " bnfell2descent_viaisog : courbe singuliere !!"); if ((!gequalgs(ell_get_a1(ell), 0) || !gequalgs(ell_get_a3(ell), 0)) || !gequalgs(ell_get_a6(ell), 0)) pari_err(user, " bnfell2descent_viaisog : la courbe n'est pas sous la forme [0,a,0,b,0]"); if ((gcmpgs(denom(algtobasis(bnf, ell_get_a2(ell))), 1) > 0) || (gcmpgs(denom(algtobasis(bnf, ell_get_a4(ell))), 1) > 0)) pari_err(user, " bnfell2descent_viaisog : coefficients non entiers"); p1 = cgetg(4, t_VEC); gel(p1, 1) = gen_1; gel(p1, 2) = gcopy(ell_get_a2(ell)); gel(p1, 3) = gcopy(ell_get_a4(ell)); P = gmul(gtopoly(p1, -1), gmodulsg(1, member_pol(bnf))); Pfact = gcopy(gel(polfnf(P, member_pol(bnf)), 1)); tors = stoi(glength(Pfact)); if (glength(Pfact) > 1) { p2 = cgetg(4, t_VEC); p3 = cgetg(4, t_VEC); gel(p3, 1) = gen_0; gel(p3, 2) = gen_0; gel(p3, 3) = gen_1; gel(p2, 1) = p3; p4 = cgetg(4, t_VEC); gel(p4, 1) = gneg(polcoeff0(gel(Pfact, 1), 0, -1)); gel(p4, 2) = gen_0; gel(p4, 3) = gen_1; gel(p2, 2) = p4; p5 = cgetg(4, t_VEC); gel(p5, 1) = gneg(polcoeff0(gel(Pfact, 2), 0, -1)); gel(p5, 2) = gen_0; gel(p5, 3) = gen_1; gel(p2, 3) = p5; pointstriv = p2; } else { p6 = cgetg(2, t_VEC); p7 = cgetg(4, t_VEC); gel(p7, 1) = gen_0; gel(p7, 2) = gen_0; gel(p7, 3) = gen_1; gel(p6, 1) = p7; pointstriv = p6; } apinit = gmulsg(-2, ell_get_a2(ell)); bpinit = gsub(gsqr(ell_get_a2(ell)), gmulsg(4, ell_get_a4(ell))); /* calcul des ideaux premiers de plist */ /* et de quelques renseignements associes */ plist = gcopy(gel(idealfactor(bnf, gmulsg(6, member_disc(ell))), 1)); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" Recherche de points triviaux sur la courbe\n"); P = gmul(P, x); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf("Y^2 = %Ps\n", P); pointstriv = concat(pointstriv, nfratpoint(member_nf(bnf), P, LIMTRIV, gen_0, prec)); if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) { pari_printf("points triviaux sur E(K) = \n"); pari_printf("%Ps\n", lift(pointstriv)); pari_printf("\n"); } KS2prod = gcopy(ell_get_a4(ell)); oddclass = gen_0; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); while (gequal0(oddclass)) { KS2gen = bnfsunit(bnf, gtrans(gel(idealfactor(bnf, KS2prod), 1)), prec); oddclass = gmodgs(gel(gel(KS2gen, 5), 1), 2); if (gequal0(oddclass)) KS2prod = idealmul(bnf, KS2prod, gel(gel(gel(KS2gen, 5), 3), 1)); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 3, &KS2gen, &oddclass, &KS2prod); } } KS2gen = gcopy(gel(KS2gen, 1)); /* A CHANGER : KS2gen = matbasistoalg(bnf,KS2gen); */ l8 = glength(KS2gen); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; for (i = 1; i <= l8; ++i) { gel(KS2gen, i) = basistoalg(bnf, gel(KS2gen, i)); if (low_stack(st_lim, stack_lim(btop, 1))) KS2gen = gerepilecopy(btop, KS2gen); } } KS2gen = concat(gmodulo(lift(gettufu(bnf)), member_pol(bnf)), KS2gen); if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) { pari_printf("#K(b,2)gen = %ld\n", glength(KS2gen)); pari_printf("K(b,2)gen = %Ps\n", KS2gen); } listpoints = nfellcount(member_nf(bnf), ell_get_a2(ell), ell_get_a4(ell), KS2gen, pointstriv, prec); pointgen = gcopy(gel(listpoints, 1)); if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) { pari_printf("points sur E(K) = %Ps\n", lift(pointgen)); pari_printf("\n"); } n1 = gcopy(gel(gel(listpoints, 2), 1)); n2 = gcopy(gel(gel(listpoints, 2), 2)); certain = stoi(gequal(n1, n2)); if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) { if (!gequal0(certain)) { pari_printf("[E(K):phi'(E'(K))] = %Ps\n", shifti(gen_1, gtos(n1))); pari_printf("#S^(phi')(E'/K) = %Ps\n", shifti(gen_1, gtos(n2))); pari_printf("#III(E'/K)[phi'] = 1\n"); pari_printf("\n"); } else { pari_printf("[E(K):phi'(E'(K))] >= %Ps\n", shifti(gen_1, gtos(n1))); pari_printf("#S^(phi')(E'/K) = %Ps\n", shifti(gen_1, gtos(n2))); pari_printf("#III(E'/K)[phi'] <= %Ps\n", shifti(gen_1, gtos(gsub(n2, n1)))); pari_printf("\n"); } } KS2prod = bpinit; oddclass = gen_0; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); while (gequal0(oddclass)) { KS2gen = bnfsunit(bnf, gtrans(gel(idealfactor(bnf, KS2prod), 1)), prec); oddclass = gmodgs(gel(gel(KS2gen, 5), 1), 2); if (gequal0(oddclass)) KS2prod = idealmul(bnf, KS2prod, gel(gel(gel(KS2gen, 5), 3), 1)); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 3, &KS2gen, &oddclass, &KS2prod); } } KS2gen = gcopy(gel(KS2gen, 1)); /* A CHANGER KS2gen = matbasistoalg(bnf,KS2gen); */ l9 = glength(KS2gen); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; for (i = 1; i <= l9; ++i) { gel(KS2gen, i) = basistoalg(bnf, gel(KS2gen, i)); if (low_stack(st_lim, stack_lim(btop, 1))) KS2gen = gerepilecopy(btop, KS2gen); } } KS2gen = concat(gmodulo(lift(gettufu(bnf)), member_pol(bnf)), KS2gen); if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) { pari_printf("#K(a^2-4b,2)gen = %ld\n", glength(KS2gen)); pari_printf("K(a^2-4b,2)gen = %Ps\n", KS2gen); } p10 = cgetg(4, t_VEC); gel(p10, 1) = gen_1; gel(p10, 2) = gcopy(apinit); gel(p10, 3) = gcopy(bpinit); P = gmul(gtopoly(p10, -1), gmodulsg(1, member_pol(bnf))); Pfact = gcopy(gel(polfnf(P, member_pol(bnf)), 1)); if (glength(Pfact) > 1) { p11 = cgetg(4, t_VEC); p12 = cgetg(4, t_VEC); gel(p12, 1) = gen_0; gel(p12, 2) = gen_0; gel(p12, 3) = gen_1; gel(p11, 1) = p12; p13 = cgetg(4, t_VEC); gel(p13, 1) = gneg(polcoeff0(gel(Pfact, 1), 0, -1)); gel(p13, 2) = gen_0; gel(p13, 3) = gen_1; gel(p11, 2) = p13; p14 = cgetg(4, t_VEC); gel(p14, 1) = gneg(polcoeff0(gel(Pfact, 2), 0, -1)); gel(p14, 2) = gen_0; gel(p14, 3) = gen_1; gel(p11, 3) = p14; pointstriv = p11; } else { p15 = cgetg(2, t_VEC); p16 = cgetg(4, t_VEC); gel(p16, 1) = gen_0; gel(p16, 2) = gen_0; gel(p16, 3) = gen_1; gel(p15, 1) = p16; pointstriv = p15; } if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" Recherche de points triviaux sur la courbe\n"); P = gmul(P, x); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf("Y^2 = %Ps\n", P); pointstriv = concat(pointstriv, nfratpoint(member_nf(bnf), P, LIMTRIV, gen_0, prec)); if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) { pari_printf("points triviaux sur E'(K) = \n"); pari_printf("%Ps\n", lift(pointstriv)); pari_printf("\n"); } listpoints = nfellcount(member_nf(bnf), apinit, bpinit, KS2gen, pointstriv, prec); if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) pari_printf("points sur E'(K) = %Ps\n", lift(gel(listpoints, 1))); np1 = gcopy(gel(gel(listpoints, 2), 1)); np2 = gcopy(gel(gel(listpoints, 2), 2)); l17 = glength(gel(listpoints, 1)); { long i; p18 = cgetg(l17+1, t_VEC); for (i = 1; i <= l17; ++i) gel(p18, i) = gen_0; } listpoints2 = p18; l19 = glength(gel(listpoints, 1)); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; GEN p21 = gen_0; /* vec */ for (i = 1; i <= l19; ++i) { p21 = cgetg(3, t_VEC); gel(p21, 1) = gen_0; gel(p21, 2) = gen_0; gel(listpoints2, i) = p21; aux1 = gsqr(gel(gel(gel(listpoints, 1), i), 1)); if (!gequalgs(aux1, 0)) { aux2 = gcopy(gel(gel(gel(listpoints, 1), i), 2)); gel(gel(listpoints2, i), 1) = gdivgs(gdiv(gsqr(aux2), aux1), 4); gel(gel(listpoints2, i), 2) = gdivgs(gdiv(gmul(aux2, gsub(bpinit, aux1)), aux1), 8); } else gel(listpoints2, i) = gcopy(gel(gel(listpoints, 1), i)); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 4, &p21, &listpoints2, &aux1, &aux2); } } if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) { pari_printf("points sur E(K) = %Ps\n", lift(listpoints2)); pari_printf("\n"); } pointgen = concat(pointgen, listpoints2); certainp = stoi(gequal(np1, np2)); if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) { if (!gequal0(certainp)) { pari_printf("[E'(K):phi(E(K))] = %Ps\n", shifti(gen_1, gtos(np1))); pari_printf("#S^(phi)(E/K) = %Ps\n", shifti(gen_1, gtos(np2))); pari_printf("#III(E/K)[phi] = 1\n"); pari_printf("\n"); } else { pari_printf("[E'(K):phi(E(K))] >= %Ps\n", shifti(gen_1, gtos(np1))); pari_printf("#S^(phi)(E/K) = %Ps\n", shifti(gen_1, gtos(np2))); pari_printf("#III(E/K)[phi] <= %Ps\n", shifti(gen_1, gtos(gsub(np2, np1)))); pari_printf("\n"); } if ((gequal0(certain)) && (gcmp(np2, np1) > 0)) pari_printf("%Ps <= ", shifti(gen_1, gtos(gsub(np2, np1)))); pari_printf("#III(E/K)[2] "); if (!gequal0(certain) && !gequal0(certainp)) pari_printf(" "); else pari_printf("<"); pari_printf("= %Ps\n", shifti(gen_1, gtos(gsub(gsub(gadd(n2, np2), n1), np1)))); pari_printf("#E(K)[2] = %Ps\n", shifti(gen_1, gtos(tors))); } rang = gsubgs(gadd(n1, np1), 2); if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) { if (!gequal0(certain) && !gequal0(certainp)) { pari_printf("#E(K)/2E(K) = %Ps\n", shifti(gen_1, gtos(gadd(rang, tors)))); pari_printf("rang = %Ps\n", rang); pari_printf("\n"); } else { pari_printf("#E(K)/2E(K) >= %Ps\n", shifti(gen_1, gtos(gadd(rang, tors)))); pari_printf("\n"); pari_printf("%Ps <= rang <= %Ps\n", rang, gsubgs(gadd(n2, np2), 2)); pari_printf("\n"); } } strange = gmodgs(gsub(gsub(gadd(n2, np2), n1), np1), 2); if (!gequal0(strange)) { if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) { pari_printf(" !!! III doit etre un carre !!!\n"); pari_printf("donc\n"); } if (!gequal0(certain)) { np1 = gaddgs(np1, 1); certainp = stoi(gequal(np1, np2)); if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) { if (!gequal0(certainp)) { pari_printf("[E'(K):phi(E(K))] = %Ps\n", shifti(gen_1, gtos(np1))); pari_printf("#S^(phi)(E/K) = %Ps\n", shifti(gen_1, gtos(np2))); pari_printf("#III(E/K)[phi] = 1\n"); pari_printf("\n"); } else { pari_printf("[E'(K):phi(E(K))] >= %Ps\n", shifti(gen_1, gtos(np1))); pari_printf("#S^(phi)(E/K) = %Ps\n", shifti(gen_1, gtos(np2))); pari_printf("#III(E/K)[phi] <= %Ps\n", shifti(gen_1, gtos(gsub(np2, np1)))); pari_printf("\n"); } } } else { if (!gequal0(certainp)) { n1 = gaddgs(n1, 1); certain = stoi(gequal(n1, n2)); if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) { if (!gequal0(certain)) { pari_printf("[E(K):phi'(E'(K))] = %Ps\n", shifti(gen_1, gtos(n1))); pari_printf("#S^(phi')(E'/K) = %Ps\n", shifti(gen_1, gtos(n2))); pari_printf("#III(E'/K)[phi'] = 1\n"); pari_printf("\n"); } else { pari_printf("[E(K):phi'(E'(K))] >= %Ps\n", shifti(gen_1, gtos(n1))); pari_printf("#S^(phi')(E'/K) = %Ps\n", shifti(gen_1, gtos(n2))); pari_printf("#III(E'/K)[phi'] <= %Ps\n", shifti(gen_1, gtos(gsub(n2, n1)))); pari_printf("\n"); } } } else n1 = gaddgs(n1, 1); } if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) { if ((gequal0(certain)) && (gcmp(np2, np1) > 0)) pari_printf("%Ps <= ", shifti(gen_1, gtos(gsub(np2, np1)))); pari_printf("#III(E/K)[2] "); if (!gequal0(certain) && !gequal0(certainp)) pari_printf(" "); else pari_printf("<"); pari_printf("= %Ps\n", shifti(gen_1, gtos(gsub(gsub(gadd(n2, np2), n1), np1)))); pari_printf("#E(K)[2] = %Ps\n", shifti(gen_1, gtos(tors))); } rang = gsubgs(gadd(n1, np1), 2); if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) { if (!gequal0(certain) && !gequal0(certainp)) { pari_printf("#E(K)/2E(K) = %Ps\n", shifti(gen_1, gtos(gadd(rang, tors)))); pari_printf("\n"); pari_printf("rang = %Ps\n", rang); pari_printf("\n"); } else { pari_printf("#E(K)/2E(K) >= %Ps\n", shifti(gen_1, gtos(gadd(rang, tors)))); pari_printf("\n"); pari_printf("%Ps <= rang <= %Ps\n", rang, gsubgs(gadd(n2, np2), 2)); pari_printf("\n"); } } } /* fin de strange */ if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) pari_printf("points = %Ps\n", pointgen); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf("fin de bnfell2descent_viaisog\n"); p20 = cgetg(4, t_VEC); gel(p20, 1) = gcopy(rang); gel(p20, 2) = gadd(gsubgs(gadd(n2, np2), 2), tors); gel(p20, 3) = gcopy(pointgen); p20 = gerepilecopy(ltop, p20); return p20; } GEN nfchinremain(GEN nf, GEN b, GEN fact) { pari_sp ltop = avma; GEN l = gen_0, fact2 = gen_0; GEN p1 = gen_0; /* vec */ /* MODI remove i */ if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("entree dans nfchinremain\n"); l = stoi(glength(gel(fact, 1))); { long i; p1 = cgetg(gtos(l)+1, t_VEC); for (i = 1; gcmpsg(i, l) <= 0; ++i) gel(p1, i) = idealdiv(nf, b, idealpow0(nf, gcoeff(fact, i, 1), gcoeff(fact, i, 2), 0)); } fact2 = p1; /* for( i = 1, l, */ /* fact2[i] = idealdiv(nf,b,idealpow(nf,fact[i,1],fact[i,2]))); */ fact2 = idealaddtoone0(nf, fact2, NULL); /* A CHANGER : fact2 = matbasistoalg(nf,fact2); */ { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN i = gen_0; for (i = gen_1; gcmp(i, l) <= 0; i = gaddgs(i, 1)) { gel(fact2, gtos(i)) = basistoalg(nf, gel(fact2, gtos(i))); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &i, &fact2); } } if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("fin de nfchinremain\n"); fact2 = gerepilecopy(ltop, fact2); return fact2; } GEN bnfqfsolve2(GEN bnf, GEN aleg, GEN bleg, GEN auto_s, long prec) /* vec */ { pari_sp ltop = avma; GEN p1 = gen_0; /* vec */ GEN y = pol_x(fetch_user_var("y")), aux = gen_0, solvepolrel = gen_0, auxsolve = gen_0, solvepolabs = gen_0, exprxy = gen_0, rrrnf = gen_0, bbbnf = gen_0, SL0 = gen_0, SL1 = gen_0, SL = gen_0, sunL = gen_0, fondsunL = gen_0, normfondsunL = gen_0, SK = gen_0, sunK = gen_0, fondsunK = gen_0, vecbleg = gen_0, matnorm = gen_0, matnormmod = gen_0, expsolution = gen_0, solution = gen_0, reste = gen_0, carre = gen_0, verif = gen_0, x0 = gen_0, x1 = gen_0, x = pol_x(fetch_user_var("x")); long l2, l3, l4; GEN p5 = gen_0; /* vec */ long l6; GEN p7 = gen_0; /* vec */ long l8, l9; GEN p10 = gen_0; /* vec */ long l11, l12; GEN p13 = gen_0; long l14; GEN p15 = gen_0; GEN p16 = gen_0; /* vec */ if (!auto_s) { p1 = cgetg(2, t_VEC); gel(p1, 1) = gcopy(y); auto_s = p1; } /* MODI remove i */ /* MODI add x0, x1 */ if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf("entree dans bnfqfsolve2\n"); solvepolrel = gsub(gsqr(x), aleg); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("aleg = %Ps\n", aleg); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("bleg = %Ps\n", bleg); if (glength(auto_s) > 1) { if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("factorisation du discriminant avec les auto_smorhpismes de bnf\n"); l2 = glength(auto_s); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; for (i = 2; i <= l2; ++i) { aux = gabs(polresultant0(gsub(lift(aleg), gsubst(lift(aleg), gvar(y), gel(auto_s, i))), member_pol(bnf), -1, 0), prec); if (!gequal0(aux)) addprimes(gel(factor(aux), 1)); if (low_stack(st_lim, stack_lim(btop, 1))) aux = gerepilecopy(btop, aux); } } } auxsolve = rnfequation0(bnf, solvepolrel, 1); solvepolabs = gcopy(gel(auxsolve, 1)); exprxy = gcopy(gel(auxsolve, 2)); if (!gequal0(gel(auxsolve, 3))) if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf(" CECI EST LE NOUVEAU CAS auxsolve[3] != 0\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" bbbnfinit %Ps\n", solvepolabs); rrrnf = rnfinit(bnf, solvepolrel); bbbnf = Buchall(solvepolabs, nf_FORCE, prec); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" done\n"); SL0 = gen_1; if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("bbbnf.clgp = %Ps\n", member_clgp(bbbnf)); l3 = glength(gel(member_clgp(bbbnf), 2)); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; for (i = 1; i <= l3; ++i) { if (gequal0(gmodgs(gel(gel(member_clgp(bbbnf), 2), i), 2))) SL0 = idealmul(bbbnf, SL0, gcoeff(gel(gel(member_clgp(bbbnf), 3), i), 1, 1)); if (low_stack(st_lim, stack_lim(btop, 1))) SL0 = gerepilecopy(btop, SL0); } } SL1 = idealmul(bbbnf, SL0, rnfelementup(rrrnf, bleg)); SL = gtrans(gel(idealfactor(bbbnf, SL1), 1)); sunL = bnfsunit(bbbnf, SL, prec); l4 = glength(gel(sunL, 1)); { long i; p5 = cgetg(l4+1, t_VEC); for (i = 1; i <= l4; ++i) gel(p5, i) = basistoalg(bbbnf, gel(gel(sunL, 1), i)); } /* A CHANGER : fondsunL = concat(bbbnf.futu,matbasistoalg(bbbnf,sunL[1])); */ fondsunL = concat(getfutu(bbbnf), p5); normfondsunL = gnorm(rnfelementabstorel(rrrnf, fondsunL)); SK = gtrans(gel(idealfactor(bnf, idealnorm(bbbnf, SL1)), 1)); sunK = bnfsunit(bnf, SK, prec); l6 = glength(gel(sunK, 1)); { long i; p7 = cgetg(l6+1, t_VEC); for (i = 1; i <= l6; ++i) gel(p7, i) = basistoalg(bnf, gel(gel(sunK, 1), i)); } /* A CHANGER : fondsunK = concat(bnf.futu,matbasistoalg(bnf,sunK[1])); */ fondsunK = concat(getfutu(bnf), p7); vecbleg = bnfissunit(bnf, sunK, bleg); l8 = glength(normfondsunL); l9 = glength(fondsunK); { long i, j; p10 = cgetg(l8+1, t_MAT); for (j = 1; j <= l8; ++j) { gel(p10, j) = cgetg(l9+1, t_COL); for (i = 1; i <= l9; ++i) gcoeff(p10, i, j) = gen_0; } } matnorm = p10; l11 = glength(normfondsunL); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; for (i = 1; i <= l11; ++i) { gel(matnorm, i) = lift(bnfissunit(bnf, sunK, gel(normfondsunL, i))); if (low_stack(st_lim, stack_lim(btop, 1))) matnorm = gerepilecopy(btop, matnorm); } } matnormmod = gmul(matnorm, gmodulss(1, 2)); expsolution = lift(inverseimage(matnormmod, gmul(vecbleg, gmodulss(1, 2)))); if (gequal(expsolution, cgetg(1, t_COL))) pari_err(user, " bnfqfsolve2 : IL N'Y A PAS DE SOLUTION "); l12 = glength(expsolution); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; p13 = gen_1; for (i = 1; i <= l12; ++i) { p13 = gmul(p13, gpow(gel(fondsunL, i), gel(expsolution, i), prec)); if (low_stack(st_lim, stack_lim(btop, 1))) p13 = gerepilecopy(btop, p13); } } solution = p13; solution = rnfelementabstorel(rrrnf, solution); reste = gdivgs(gsub(lift(vecbleg), gmul(matnorm, expsolution)), 2); l14 = glength(vecbleg); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; p15 = gen_1; for (i = 1; i <= l14; ++i) { p15 = gmul(p15, gpow(gel(fondsunK, i), gel(reste, i), prec)); if (low_stack(st_lim, stack_lim(btop, 1))) p15 = gerepilecopy(btop, p15); } } carre = p15; solution = gmul(solution, carre); x1 = polcoeff0(lift(solution), 1, gvar(x)); x0 = polcoeff0(lift(solution), 0, gvar(x)); verif = gsub(gsub(gsqr(x0), gmul(aleg, gsqr(x1))), bleg); if (!gequal0(verif)) pari_err(user, " bnfqfsolve2 : MAUVAIS POINT"); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf("fin de bnfqfsolve2\n"); p16 = cgetg(4, t_VEC); gel(p16, 1) = gcopy(x0); gel(p16, 2) = gcopy(x1); gel(p16, 3) = gen_1; p16 = gerepilecopy(ltop, p16); return p16; } GEN bnfqfsolve(GEN bnf, GEN aleg, GEN bleg, GEN flag3, GEN auto_s, long prec) { pari_sp ltop = avma; GEN p1 = gen_0; /* vec */ GEN y = pol_x(fetch_user_var("y")), nf = gen_0, aa = gen_0, bb = gen_0, na = gen_0, nb = gen_0, maxnb = gen_0, mat = gen_0, resl = gen_0, t = gen_0, sq = gen_0, pol = gen_0, vecrat = gen_0, alpha = gen_0, xx = gen_0, yy = gen_0, borne = gen_0, test = gen_0, sun = gen_0, fact = gen_0, suni = gen_0, f = gen_0, l = gen_0, aux = gen_0, alpha2 = gen_0, maxnbiter = gen_0, idbb = gen_0, rem = gen_0, nbiter = gen_0, mask = gen_0, oldnb = gen_0, newnb = gen_0, bor = gen_0, testici = gen_0, de = gen_0, xxp = gen_0, yyp = gen_0, rap = gen_0, verif = gen_0; GEN p2 = gen_0, p3 = gen_0; /* vec */ GEN x = pol_x(fetch_user_var("x")); if (!auto_s) { p1 = cgetg(2, t_VEC); gel(p1, 1) = gcopy(y); auto_s = p1; } /* MODI remove k, maxna */ if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("entree dans bnfqfsolve\n"); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf("(a,b) = (%Ps,%Ps)\n", aleg, bleg); nf = gcopy(member_nf(bnf)); aleg = gmodulo(lift(aleg), member_pol(nf)); aa = gcopy(aleg); bleg = gmodulo(lift(bleg), member_pol(nf)); bb = gcopy(bleg); if (gequal0(aa)) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de bnfqfsolve\n"); p2 = cgetg(4, t_COL); gel(p2, 1) = gen_0; gel(p2, 2) = gen_1; gel(p2, 3) = gen_0; p2 = gerepilecopy(ltop, p2); return p2; } if (gequal0(bb)) { if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("fin de bnfqfsolve\n"); p3 = cgetg(4, t_COL); gel(p3, 1) = gen_0; gel(p3, 2) = gen_0; gel(p3, 3) = gen_1; p3 = gerepilecopy(ltop, p3); return p3; } na = gabs(gnorm(aa), prec); nb = gabs(gnorm(bb), prec); if (gcmp(na, nb) > 0) maxnb = na; else maxnb = nb; maxnb = gshift(maxnb, 20); mat = gmodulo(matid(3), member_pol(nf)); borne = gen_1; test = gen_0; nbiter = gen_0; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN p4 = gen_0, p5 = gen_0, p6 = gen_0, p7 = gen_0, p8 = gen_0, p9 = gen_0, p10 = gen_0, p11 = gen_0, p12 = gen_0, p13 = gen_0; /* vec */ long l14; GEN p15 = gen_0; /* vec */ long l16; GEN p17 = gen_0, p18 = gen_0; /* vec */ while (1) { if (!gequal0(flag3) && (gcmpgs(gel(member_clgp(bnf), 1), 1) > 0)) { resl = gtrans(bnfqfsolve2(bnf, aa, bb, auto_s, prec)); break; } if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) { p4 = cgetg(7, t_VEC); gel(p4, 1) = gcopy(na); gel(p4, 2) = gcopy(nb); gel(p4, 3) = gcopy(aa); gel(p4, 4) = gcopy(bb); gel(p4, 5) = gnorm(aa); gel(p4, 6) = gnorm(bb); pari_printf("(na,nb,a,b) = %Ps\n", lift(p4)); } if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("***%Ps*** \n", nb); if (gcmp(nb, maxnb) >= 0) { mat = gmodulo(matid(3), member_pol(nf)); aa = gcopy(aleg); bb = gcopy(bleg); na = gabs(gnorm(aleg), prec); nb = gabs(gnorm(bleg), prec); } if (gequal1(aa)) { p5 = cgetg(4, t_COL); gel(p5, 1) = gen_1; gel(p5, 2) = gen_1; gel(p5, 3) = gen_0; resl = p5; break; } if (gequal1(bb)) { p6 = cgetg(4, t_COL); gel(p6, 1) = gen_1; gel(p6, 2) = gen_0; gel(p6, 3) = gen_1; resl = p6; break; } if (gequal1(gadd(aa, bb))) { p7 = cgetg(4, t_COL); gel(p7, 1) = gen_1; gel(p7, 2) = gen_1; gel(p7, 3) = gen_1; resl = p7; break; } if (gequal0(gadd(aa, bb))) { p8 = cgetg(4, t_COL); gel(p8, 1) = gen_0; gel(p8, 2) = gen_1; gel(p8, 3) = gen_1; resl = p8; break; } if (gequal(aa, bb) && !gequalgs(aa, 1)) { t = gmul(aa, gel(mat, 1)); gel(mat, 1) = gcopy(gel(mat, 3)); gel(mat, 3) = gcopy(t); aa = gen_m1; na = gen_1; } if (!gequal0(gissquare(na))) { sq = nfsqrt(nf, aa, prec); if (!gequal(sq, cgetg(1, t_VEC))) { p9 = cgetg(4, t_COL); gel(p9, 1) = gcopy(gel(sq, 1)); gel(p9, 2) = gen_1; gel(p9, 3) = gen_0; resl = p9; break; } } if (!gequal0(gissquare(nb))) { sq = nfsqrt(nf, bb, prec); if (!gequal(sq, cgetg(1, t_VEC))) { p10 = cgetg(4, t_COL); gel(p10, 1) = gcopy(gel(sq, 1)); gel(p10, 2) = gen_0; gel(p10, 3) = gen_1; resl = p10; break; } } if (gcmp(na, nb) > 0) { t = aa; aa = bb; bb = t; t = na; na = nb; nb = t; t = gcopy(gel(mat, 3)); gel(mat, 3) = gcopy(gel(mat, 2)); gel(mat, 2) = gcopy(t); } if (gequal1(nb)) { if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) { p11 = cgetg(3, t_VEC); gel(p11, 1) = gcopy(aa); gel(p11, 2) = gcopy(bb); pari_printf("(a,b) = %Ps\n", lift(p11)); } if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) { p12 = cgetg(3, t_VEC); gel(p12, 1) = gcopy(na); gel(p12, 2) = gcopy(nb); pari_printf("(na,nb) = %Ps\n", lift(p12)); } if (gequal(aleg, aa) && gequal(bleg, bb)) mat = gmodulo(matid(3), member_pol(nf)); if (!gequal0(flag3)) { resl = gtrans(bnfqfsolve2(bnf, aa, bb, auto_s, prec)); break; } pol = gadd(gmul(aa, gsqr(x)), bb); vecrat = nfratpoint(nf, pol, borne = gaddgs(borne, 1), gen_1, prec); if (!gequalgs(vecrat, 0)) { p13 = cgetg(4, t_COL); gel(p13, 1) = gcopy(gel(vecrat, 2)); gel(p13, 2) = gcopy(gel(vecrat, 1)); gel(p13, 3) = gcopy(gel(vecrat, 3)); resl = p13; break; } alpha = gen_0; if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("borne = %Ps\n", borne); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); while (gequal0(alpha)) { xx = nfrandint(nf, borne); yy = nfrandint(nf, borne); borne = gaddgs(borne, 1); alpha = gsub(gsqr(xx), gmul(aa, gsqr(yy))); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 4, &xx, &yy, &borne, &alpha); } } bb = gmul(bb, alpha); nb = gmul(nb, gabs(gnorm(alpha), prec)); t = gadd(gmul(xx, gel(mat, 1)), gmul(yy, gel(mat, 2))); gel(mat, 2) = gadd(gmul(xx, gel(mat, 2)), gmul(gmul(aa, yy), gel(mat, 1))); gel(mat, 1) = gcopy(t); gel(mat, 3) = gmul(gel(mat, 3), alpha); } else { test = gen_1; if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("on factorise bb = %Ps\n", bb); sun = bnfsunit(bnf, gtrans(gel(idealfactor(bnf, bb), 1)), prec); fact = lift(bnfissunit(bnf, sun, bb)); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("fact = %Ps\n", fact); l14 = glength(gel(sun, 1)); { long i; p15 = cgetg(l14+1, t_VEC); for (i = 1; i <= l14; ++i) gel(p15, i) = basistoalg(bnf, gel(gel(sun, 1), i)); } suni = concat(getfutu(bnf), p15); l16 = glength(suni); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; for (i = 1; i <= l16; ++i) { if (!gequal0(f = gshift(gel(fact, i), -1))) { test = gen_0; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long k; for (k = 1; k <= 3; ++k) { gcoeff(mat, k, 3) = gdiv(gcoeff(mat, k, 3), gpow(gel(suni, i), f, prec)); if (low_stack(st_lim, stack_lim(btop, 1))) mat = gerepilecopy(btop, mat); } } nb = gdiv(nb, gpow(gabs(gnorm(gel(suni, i)), prec), gmulsg(2, f), prec)); bb = gdiv(bb, gpow(gel(suni, i), gmulsg(2, f), prec)); } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 5, &f, &test, &mat, &nb, &bb); } } if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("on factorise bb = %Ps\n", bb); fact = idealfactor(nf, bb); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("fact = %Ps\n", fact); l = stoi(glength(gel(fact, 1))); if (!gequal0(test)) { aux = gen_1; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN i = gen_0; for (i = gen_1; gcmp(i, l) <= 0; i = gaddgs(i, 1)) { if ((!gequal0(f = gshift(gcoeff(fact, gtos(i), 2), -1)) && (gequal0(gmodgs(gel(gcoeff(fact, gtos(i), 1), 1), 2)))) && (!nfpsquareodd(nf, aa, gcoeff(fact, gtos(i), 1), prec))) aux = idealmul(nf, aux, idealpow0(nf, gcoeff(fact, gtos(i), 1), f, 0)); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 3, &i, &f, &aux); } } if (!gequalgs(aux, 1)) { test = gen_0; alpha = basistoalg(nf, idealappr0(nf, idealinv(nf, aux), 0)); alpha2 = gsqr(alpha); bb = gmul(bb, alpha2); nb = gmul(nb, gabs(gnorm(alpha2), prec)); gel(mat, 3) = gmul(gel(mat, 3), alpha); } } if (!gequal0(test)) { maxnbiter = shifti(gen_1, gtos(l)); { long i; p17 = cgetg(gtos(l)+1, t_VEC); for (i = 1; gcmpsg(i, l) <= 0; ++i) gel(p17, i) = nfissquarep(nf, aa, gcoeff(fact, i, 1), gcoeff(fact, i, 2), prec); } sq = p17; l = stoi(glength(sq)); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) { pari_printf("sq = %Ps\n", sq); pari_printf("fact = %Ps\n", fact); pari_printf("l = %Ps\n", l); } if (gcmpgs(l, 1) > 0) { idbb = idealhnf0(nf, bb, NULL); rem = nfchinremain(nf, idbb, fact); } test = gen_1; nbiter = gen_1; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN p19 = gen_0, p20 = gen_0, p21 = gen_0; /* vec */ while (!gequal0(test) && (gcmp(nbiter, maxnbiter) <= 0)) { if (gcmpgs(l, 1) > 0) { mask = nbiter; xx = gen_0; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN i = gen_0; for (i = gen_1; gcmp(i, l) <= 0; i = gaddgs(i, 1)) { if (!gequal0(gmodgs(mask, 2))) xx = gadd(xx, gmul(gel(rem, gtos(i)), gel(sq, gtos(i)))); else xx = gsub(xx, gmul(gel(rem, gtos(i)), gel(sq, gtos(i)))); mask = gshift(mask, -1); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 3, &i, &xx, &mask); } } } else { test = gen_0; xx = gcopy(gel(sq, 1)); } xx = mynfeltmod(nf, xx, bb); alpha = gsub(gsqr(xx), aa); if (gequal0(alpha)) { p19 = cgetg(4, t_COL); gel(p19, 1) = gcopy(xx); gel(p19, 2) = gen_1; gel(p19, 3) = gen_0; resl = p19; goto label6; } t = gdiv(alpha, bb); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) { p20 = cgetg(3, t_VEC); gel(p20, 1) = gcopy(alpha); gel(p20, 2) = gcopy(bb); pari_printf("[alpha,bb] = %Ps\n", p20); } oldnb = nb; newnb = gabs(gnorm(t), prec); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) { p21 = cgetg(4, t_VEC); gel(p21, 1) = gcopy(oldnb); gel(p21, 2) = gcopy(newnb); gel(p21, 3) = gadd(gdiv(oldnb, newnb), real_0(prec)); pari_printf("[oldnb,newnb,oldnb/newnb] = %Ps\n", p21); } { pari_sp btop = avma, st_lim = stack_lim(btop, 1); while (gcmp(nb, newnb) > 0) { gel(mat, 3) = gmul(gel(mat, 3), t); bb = t; nb = newnb; t = gadd(gmul(xx, gel(mat, 1)), gel(mat, 2)); gel(mat, 2) = gadd(gmul(aa, gel(mat, 1)), gmul(xx, gel(mat, 2))); gel(mat, 1) = gcopy(t); xx = mynfeltmod(nf, gneg(xx), bb); alpha = gsub(gsqr(xx), aa); t = gdiv(alpha, bb); newnb = gabs(gnorm(t), prec); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 7, &mat, &bb, &nb, &t, &xx, &alpha, &newnb); } } if (gequal(nb, oldnb)) nbiter = gaddgs(nbiter, 1); else test = gen_0; if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 15, &mask, &xx, &test, &alpha, &p19, &resl, &t, &p20, &oldnb, &newnb, &p21, &mat, &bb, &nb, &nbiter); } } if (gequal(nb, oldnb)) { if (!gequal0(flag3)) { resl = gtrans(bnfqfsolve2(bnf, aa, bb, auto_s, prec)); break; } pol = gadd(gmul(aa, gsqr(x)), bb); vecrat = nfratpoint(nf, pol, gshift(borne = gaddgs(borne, 1), 1), gen_1, prec); if (!gequalgs(vecrat, 0)) { p18 = cgetg(4, t_COL); gel(p18, 1) = gcopy(gel(vecrat, 2)); gel(p18, 2) = gcopy(gel(vecrat, 1)); gel(p18, 3) = gcopy(gel(vecrat, 3)); resl = p18; break; } bor = stoi(1000); yy = gen_1; testici = gen_1; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i, l22; GEN p23 = gen_0; /* vec */ for (i = 1; i <= 10000; ++i) { l22 = degree(member_pol(nf)); { long j; p23 = cgetg(l22+1, t_COL); for (j = 1; j <= l22; ++j) gel(p23, j) = genrand(bor); } de = basistoalg(nf, p23); if (!gequal(idealadd(bnf, de, bb), matid(degree(member_pol(bnf))))) continue; xxp = mynfeltmod(bnf, gmul(de, xx), bb); yyp = mynfeltmod(bnf, gmul(de, yy), bb); rap = gadd(gdiv(gnorm(gsub(gsqr(xxp), gmul(aa, gsqr(yyp)))), gsqr(nb)), real_0(prec)); if (gcmpgs(gabs(rap, prec), 1) < 0) { if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("********** \n \n MIRACLE %Ps \n \n ***\n", rap); t = gdiv(gsub(gsqr(xxp), gmul(aa, gsqr(yyp))), bb); gel(mat, 3) = gmul(gel(mat, 3), t); bb = t; nb = gabs(gnorm(bb), prec); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("newnb = %Ps\n", nb); t = gadd(gmul(xxp, gel(mat, 1)), gmul(yyp, gel(mat, 2))); gel(mat, 2) = gadd(gmul(gmul(aa, yyp), gel(mat, 1)), gmul(xxp, gel(mat, 2))); gel(mat, 1) = gcopy(t); xx = xxp; yy = gneg(yyp); testici = gen_0; } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 12, &p23, &de, &xxp, &yyp, &rap, &t, &mat, &bb, &nb, &xx, &yy, &testici); } } if (!gequal0(testici)) { alpha = gen_0; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); while (gequal0(alpha)) { xx = nfrandint(nf, gmulsg(4, borne)); yy = nfrandint(nf, gmulsg(4, borne)); borne = gaddgs(borne, 1); alpha = gsub(gsqr(xx), gmul(aa, gsqr(yy))); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 4, &xx, &yy, &borne, &alpha); } } bb = gmul(bb, alpha); nb = gmul(nb, gabs(gnorm(alpha), prec)); t = gadd(gmul(xx, gel(mat, 1)), gmul(yy, gel(mat, 2))); gel(mat, 2) = gadd(gmul(xx, gel(mat, 2)), gmul(gmul(aa, yy), gel(mat, 1))); gel(mat, 1) = gcopy(t); gel(mat, 3) = gmul(gel(mat, 3), alpha); } } } } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 48, &resl, &p4, &mat, &aa, &bb, &na, &nb, &p5, &p6, &p7, &p8, &t, &sq, &p9, &p10, &p11, &p12, &pol, &borne, &vecrat, &p13, &alpha, &xx, &yy, &test, &sun, &fact, &p15, &suni, &f, &l, &aux, &alpha2, &maxnbiter, &p17, &idbb, &rem, &nbiter, &mask, &oldnb, &newnb, &p18, &bor, &testici, &de, &xxp, &yyp, &rap); } label6:; } resl = lift(gmul(mat, resl)); if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("resl1 = %Ps\n", resl); if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("content = %Ps\n", content(resl)); resl = gdiv(resl, content(resl)); resl = gmodulo(lift(resl), member_pol(nf)); if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("resl3 = %Ps\n", resl); fact = idealadd(nf, idealadd(nf, gel(resl, 1), gel(resl, 2)), gel(resl, 3)); fact = bnfisprincipal0(bnf, fact, 3); resl = gmul(resl, ginv(basistoalg(nf, gel(fact, 2)))); if (gcmpgs(DEBUGLEVEL_ell, 5) >= 0) pari_printf("resl4 = %Ps\n", resl); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf("resl = %Ps\n", resl); verif = stoi(gequal0(gsub(gsub(gsqr(gel(resl, 1)), gmul(aleg, gsqr(gel(resl, 2)))), gmul(bleg, gsqr(gel(resl, 3)))))); if ((gequal0(verif)) && (gcmpgs(DEBUGLEVEL_ell, 0) >= 0)) pari_err(user, " bnfqfsolve : MAUVAIS POINT"); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf("fin de bnfqfsolve\n"); resl = gerepilecopy(ltop, resl); return resl; } GEN bnfredquartique2(GEN bnf, GEN pol, GEN r, GEN a, GEN b) /* vec */ { pari_sp ltop = avma; GEN gcc = gen_0, princ = gen_0, rp = gen_0, pol2 = gen_0; GEN p1 = gen_0, p2 = gen_0; /* vec */ GEN x = pol_x(fetch_user_var("x")); GEN p3 = gen_0; /* vec */ /* MODI remove ap, den */ if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("entree dans bnfredquartique2\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) { p1 = cgetg(4, t_VEC); gel(p1, 1) = gcopy(r); gel(p1, 2) = gcopy(a); gel(p1, 3) = gcopy(b); pari_printf("%Ps\n", p1); } if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" reduction de la quartique %Ps\n", pol); if (gequal0(a)) rp = gen_0; else { gcc = idealadd(bnf, b, a); if (gequal1(gcc)) { rp = gdiv(basistoalg(bnf, gel(idealaddtoone0(member_nf(bnf), a, b), 1)), a); rp = mynfeltmod(bnf, gmul(r, rp), b); } else { princ = bnfisprincipal0(bnf, gcc, 3); if (gequal0(gel(princ, 1))) gcc = basistoalg(bnf, gel(princ, 2)); else { if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" quartique non reduite\n"); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("fin de bnfredquartique2\n"); p2 = cgetg(4, t_VEC); gel(p2, 1) = gcopy(pol); gel(p2, 2) = gen_0; gel(p2, 3) = gen_1; p2 = gerepilecopy(ltop, p2); return p2; } rp = gdiv(basistoalg(bnf, gel(idealaddtoone0(member_nf(bnf), gdiv(a, gcc), gdiv(b, gcc)), 1)), gdiv(a, gcc)); rp = gdiv(mynfeltmod(bnf, gmul(r, rp), b), gcc); b = gdiv(b, gcc); } } pol2 = gdiv(gsubst(gdiv(pol, b), gvar(x), gadd(rp, gmul(b, x))), gpowgs(b, 3)); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" quartique reduite = %Ps\n", pol2); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("fin de bnfredquartique2\n"); p3 = cgetg(4, t_VEC); gel(p3, 1) = gcopy(pol2); gel(p3, 2) = gcopy(rp); gel(p3, 3) = gcopy(b); p3 = gerepilecopy(ltop, p3); return p3; } GEN bnfell2descent_gen(GEN bnf, GEN ell, GEN ext, GEN help, GEN bigflag, GEN flag3, GEN auto_s, long prec) /* vec */ { pari_sp ltop = avma; GEN p1 = gen_0; /* vec */ GEN y = pol_x(fetch_user_var("y")), nf = gen_0, unnf = gen_0, ellnf = gen_0, A = gen_0, B = gen_0, C = gen_0, S = gen_0, plist = gen_0, Lrnf = gen_0, SLprod = gen_0, oddclass = gen_0, SLlist = gen_0, LS2gen = gen_0, polrel = gen_0, alpha = gen_0, ttheta = gen_0, KS2gen = gen_0, LS2genunit = gen_0, normcoord = gen_0, LS2coordtilda = gen_0, LS2tilda = gen_0, aux = gen_0, listgen = gen_0, listpoints = gen_0, listpointstriv = gen_0, listpointsmwr = gen_0, list = gen_0, m1 = gen_0, m2 = gen_0, loc = gen_0, lastloc = gen_0, maskwhile = gen_0, iwhile = gen_0, zc = gen_0, iaux = gen_0, liftzc = gen_0, ispointtriv = gen_0, point = gen_0, c = gen_0, b = gen_0, a = gen_0, sol = gen_0, found = gen_0, alphac = gen_0, r = gen_0, denc = gen_0, dena = gen_0, cp = gen_0, alphacp = gen_0, beta = gen_0, mattr = gen_0, vec = gen_0, z1 = gen_0, ff = gen_0, cont = gen_0, d = gen_0, e = gen_0, polorig = gen_0, pol = gen_0, redq = gen_0, transl = gen_0, multip = gen_0, UVW = gen_0, pointxx = gen_0, point2 = gen_0, v = gen_0, rang = gen_0, normLS2gen = gen_0, listELS = gen_0, listnotELS = gen_0, listlistELS = gen_0, x = pol_x(fetch_user_var("x")); long l2; GEN p3 = gen_0; /* vec */ long l4, l5; GEN p6 = gen_0; /* vec */ long l7; long l8; /* lg */ GEN p9 = gen_0; /* vec */ long l10; /* lg */ GEN p11 = gen_0, p12 = gen_0, p13 = gen_0; /* vec */ long l14; GEN p15 = gen_0; /* vec */ if (!help) help = cgetg(1, t_VEC); if (!bigflag) bigflag = gen_1; if (!flag3) flag3 = gen_1; if (!auto_s) { p1 = cgetg(2, t_VEC); gel(p1, 1) = gcopy(y); auto_s = p1; } /* MODI remove i,j,normLS2, add normLS2gen */ /* MODI add listELS,listnotELS,listlistELS */ if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("entree dans bnfell2descent_gen\n"); /* \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ */ /* construction de L(S,2) \\ */ /* \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ */ nf = gcopy(member_nf(bnf)); unnf = gmodulsg(1, member_pol(nf)); ellnf = gmul(ell, unnf); if (glength(ellnf) <= 5) ellnf = smallellinit(ellnf); A = gcopy(ell_get_a2(ellnf)); if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf("A = %Ps\n", A); B = gcopy(ell_get_a4(ellnf)); if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf("B = %Ps\n", B); C = gcopy(ell_get_a6(ellnf)); if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf("C = %Ps\n", C); S = gmulsg(6, lift(member_disc(ellnf))); plist = gcopy(gel(idealfactor(nf, S), 1)); Lrnf = gcopy(gel(ext, 3)); SLprod = gsubst(lift(deriv(gel(ext, 1),-1)), gvar(y), lift(gel(gel(ext, 2), 2))); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf("%Ps\n", gel(ext, 2)); oddclass = gen_0; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); while (gequal0(oddclass)) { /* Constructoin de S: */ SLlist = gtrans(gel(idealfactor(Lrnf, SLprod), 1)); /* Construction des S-unites */ LS2gen = bnfsunit(Lrnf, SLlist, prec); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("LS2gen = %Ps\n", LS2gen); /* on ajoute la partie paire du groupe de classes. */ oddclass = gmodgs(gel(gel(LS2gen, 5), 1), 2); if (gequal0(oddclass)) { if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf("2-class group %Ps\n", gcoeff(gel(gel(gel(LS2gen, 5), 3), 1), 1, 1)); S = gmul(S, gcoeff(gel(gel(gel(LS2gen, 5), 3), 1), 1, 1)); SLprod = idealmul(Lrnf, SLprod, gel(gel(gel(LS2gen, 5), 3), 1)); } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 5, &SLlist, &LS2gen, &oddclass, &S, &SLprod); } } polrel = gcopy(gel(ext, 1)); alpha = gmodulo(gmodulo(y, member_pol(nf)), polrel); /* alpha est l'element primitif de K */ ttheta = gmodulo(x, polrel); /* ttheta est la racine de P(x) */ KS2gen = bnfsunit(bnf, gtrans(gel(idealfactor(nf, S), 1)), prec); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf("#KS2gen = %ld\n", glength(gel(KS2gen, 1))); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf("KS2gen = %Ps\n", gel(KS2gen, 1)); LS2genunit = lift(getfutu(Lrnf)); l2 = glength(gel(LS2gen, 1)); { long i; p3 = cgetg(l2+1, t_VEC); for (i = 1; i <= l2; ++i) gel(p3, i) = lift(basistoalg(Lrnf, gel(gel(LS2gen, 1), i))); } /* A CHANGER : LS2genunit = concat(LS2genunit,lift(matbasistoalg(Lrnf,LS2gen[1]))); */ LS2genunit = concat(LS2genunit, p3); LS2genunit = gsubst(LS2genunit, gvar(x), ttheta); LS2genunit = gmul(LS2genunit, gmodulsg(1, polrel)); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf("#LS2genunit = %ld\n", glength(LS2genunit)); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf("LS2genunit = %Ps\n", LS2genunit); /* dans LS2gen, on ne garde que ceux dont la norme est un carre. */ normLS2gen = gnorm(LS2genunit); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("normLS2gen = %Ps\n", normLS2gen); l4 = glength(normLS2gen); l5 = (glength(gel(KS2gen, 1)) + glength(gel(gel(bnf, 8), 5))) + 1; { long i, j; p6 = cgetg(l4+1, t_MAT); for (j = 1; j <= l4; ++j) { gel(p6, j) = cgetg(l5+1, t_COL); for (i = 1; i <= l5; ++i) gcoeff(p6, i, j) = gen_0; } } /* matrice de l'application norme */ normcoord = p6; l7 = glength(normLS2gen); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; for (i = 1; i <= l7; ++i) { gel(normcoord, i) = bnfissunit(bnf, KS2gen, gel(normLS2gen, i)); if (low_stack(st_lim, stack_lim(btop, 1))) normcoord = gerepilecopy(btop, normcoord); } } if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("normcoord = %Ps\n", normcoord); /* construction du noyau de la norme */ LS2coordtilda = lift(matker0(gmul(normcoord, gmodulss(1, 2)), 0)); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("LS2coordtilda = %Ps\n", LS2coordtilda); l8 = lg(rowcopy(LS2coordtilda, 1)); { long i; p9 = cgetg(l8, t_VEC); for (i = 1; i < l8; ++i) gel(p9, i) = gen_0; } LS2tilda = p9; l10 = lg(rowcopy(LS2coordtilda, 1)); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i, l16; for (i = 1; i < l10; ++i) { aux = gen_1; l16 = glength(gel(LS2coordtilda, i)); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long j; for (j = 1; j <= l16; ++j) { if (gsigne(gcoeff(LS2coordtilda, j, i))) aux = gmul(aux, gel(LS2genunit, j)); if (low_stack(st_lim, stack_lim(btop, 1))) aux = gerepilecopy(btop, aux); } } gel(LS2tilda, i) = gcopy(aux); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &aux, &LS2tilda); } } if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf("LS2tilda = %Ps\n", LS2tilda); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf("norm(LS2tilda) = %Ps\n", gnorm(LS2tilda)); /* Fin de la construction de L(S,2) */ listgen = gcopy(LS2tilda); if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf("LS2gen = %Ps\n", listgen); if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf("#LS2gen = %ld\n", glength(listgen)); listpoints = cgetg(1, t_VEC); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) { p11 = cgetg(4, t_VEC); gel(p11, 1) = gcopy(A); gel(p11, 2) = gcopy(B); gel(p11, 3) = gcopy(C); pari_printf("(A,B,C) = %Ps\n", p11); } /* \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ */ /* Recherche de points triviaux \\ */ /* \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ */ if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" Recherche de points triviaux sur la courbe \n"); listpointstriv = nfratpoint(nf, gadd(gadd(gadd(gpowgs(x, 3), gmul(A, gsqr(x))), gmul(B, x)), C), LIMTRIV, gen_0, prec); listpointstriv = concat(help, listpointstriv); if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) pari_printf("points triviaux sur la courbe = %Ps\n", listpointstriv); /* \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ */ /* parcours de L(S,2) \\ */ /* \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ */ listpointsmwr = cgetg(1, t_VEC); p12 = cgetg(4, t_VEC); gel(p12, 1) = stoi(6); gel(p12, 2) = gcopy(member_disc(ellnf)); gel(p12, 3) = gen_0; list = p12; m1 = gen_0; m2 = gen_0; lastloc = gen_m1; maskwhile = shifti(gen_1, glength(listgen)); p13 = cgetg(2, t_VEC); gel(p13, 1) = gen_0; listELS = p13; listnotELS = cgetg(1, t_VEC); iwhile = gen_1; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long l17, l18, l19; GEN p20 = gen_0; /* vec */ long l21; GEN p22 = gen_0; long l23; GEN p24 = gen_0; /* vec */ long l25; GEN p26 = gen_0, p27 = gen_0, p28 = gen_0, p29 = gen_0, p30 = gen_0, p31 = gen_0, p32 = gen_0, p33 = gen_0, p34 = gen_0, p35 = gen_0, p36 = gen_0; /* vec */ while (gcmp(iwhile, maskwhile) < 0) { if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) { pari_printf("iwhile = %Ps\n", iwhile); pari_printf("listgen = %Ps\n", listgen); } /* utilise la structure de groupe pour detecter une eventuelle solubilite locale. */ if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) { pari_printf("listELS = %Ps\n", listELS); pari_printf("listnotELS = %Ps\n", listnotELS); } sol = gen_1; loc = gen_0; l17 = glength(listELS); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i, l37; for (i = 1; i <= l17; ++i) { l37 = glength(listnotELS); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long j; GEN p38 = gen_0; /* vec */ for (j = 1; j <= l37; ++j) { if (gequal(gbitxor(gel(listELS, i), gel(listnotELS, j)), iwhile)) { if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" Non ELS d'apres la structure de groupe\n"); p38 = cgetg(2, t_VEC); gel(p38, 1) = gcopy(iwhile); listnotELS = concat(listnotELS, p38); sol = gen_0; goto label7; } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 3, &p38, &listnotELS, &sol); } } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &listnotELS, &sol); } label7:; } if (gequal0(sol)) { iwhile = gaddgs(iwhile, 1); continue; } l18 = glength(listELS); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i, l39, l40; for (i = 1; i <= l18; ++i) { l39 = i + 1; l40 = glength(listELS); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); GEN j = gen_0; GEN p41 = gen_0; /* vec */ for (j = stoi(l39); gcmpgs(j, l40) <= 0; j = gaddgs(j, 1)) { if (gequal(gbitxor(gel(listELS, i), gel(listELS, gtos(j))), iwhile)) { if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" ELS d'aptres la structure de \n"); p41 = cgetg(2, t_VEC); gel(p41, 1) = gcopy(iwhile); listELS = concat(listELS, p41); loc = gen_2; goto label8; } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 4, &j, &p41, &listELS, &loc); } } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 2, &listELS, &loc); } label8:; } l19 = glength(listgen); { long i; p20 = cgetg(l19+1, t_VEC); for (i = 1; i <= l19; ++i) gel(p20, i) = gbittest(iwhile, i - 1); } iaux = gtrans(p20); iaux = gmodgs(gmul(LS2coordtilda, iaux), 2); l21 = glength(LS2genunit); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; p22 = gen_1; for (i = 1; i <= l21; ++i) { p22 = gmul(p22, gpow(gel(LS2genunit, i), gel(iaux, i), prec)); if (low_stack(st_lim, stack_lim(btop, 1))) p22 = gerepilecopy(btop, p22); } } zc = gmul(unnf, p22); if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf("zc = %Ps\n", zc); liftzc = lift(zc); /* Est-ce un point trivial ? */ ispointtriv = gen_0; l23 = glength(listpointstriv); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; GEN p42 = gen_0, p43 = gen_0; /* vec */ for (i = 1; i <= l23; ++i) { point = gcopy(gel(listpointstriv, i)); if ((glength(point) == 2) || !gequalgs(gel(point, 3), 0)) if (nfissquare(member_nf(Lrnf), gsubst(gmul(gsub(lift(gel(point, 1)), x), lift(liftzc)), gvar(y), lift(gel(gel(ext, 2), 2))), prec)) { if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" vient du point trivial %Ps\n", point); p42 = cgetg(2, t_VEC); gel(p42, 1) = gcopy(point); listpointsmwr = concat(listpointsmwr, p42); m1 = gaddgs(m1, 1); p43 = cgetg(2, t_VEC); gel(p43, 1) = gcopy(iwhile); listELS = concat(listELS, p43); if (gcmp(degre(iwhile), lastloc) > 0) m2 = gaddgs(m2, 1); sol = found = ispointtriv = gen_1; break; } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 10, &point, &p42, &listpointsmwr, &m1, &p43, &listELS, &m2, &ispointtriv, &found, &sol); } } /* Ce n'est pas un point trivial */ if (gequal0(ispointtriv)) { c = polcoeff0(liftzc, 2, -1); b = gneg(polcoeff0(liftzc, 1, -1)); a = polcoeff0(liftzc, 0, -1); sol = gen_0; found = gen_0; /* \\\\\\\\\\\\\ */ /* On cherche a ecrire zc sous la forme a-b*theta */ /* \\\\\\\\\\\\\ */ if (gequal0(c)) sol = gen_1; else { alphac = gadd(gmul(gsub(gadd(gmul(A, b), gmul(B, c)), a), c), gsqr(b)); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf("alphac = %Ps\n", alphac); r = gcopy(gel(nfsqrt(nf, gnorm(zc), prec), 1)); if (gequal0(alphac)) { /* cas particulier */ if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" on continue avec 1/zc\n"); sol = gen_1; zc = gmul(gnorm(zc), ginv(zc)); if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" zc = %Ps\n", zc); } else { /* Il faut resoudre une forme quadratique */ /* Existence (locale = globale) d'une solution : */ denc = deno(lift(c)); if (!gequalgs(denc, 1)) cp = gmul(c, gsqr(denc)); else cp = c; dena = deno(lift(alphac)); if (!gequalgs(dena, 1)) alphacp = gmul(alphac, gsqr(dena)); else alphacp = alphac; if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" symbole de Hilbert (%Ps,%Ps) = ", alphacp, cp); sol = stoi(!gequal0(loc) || (mynfhilbert(nf, alphacp, cp, prec) + 1)); if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf("%Ps\n", gsubgs(sol, 1)); if (!gequal0(sol)) { beta = gadd(gsub(gmul(A, gadd(gadd(gmul(gmul(A, b), c), gmul(B, gsqr(c))), gsqr(b))), gmul(C, gsqr(c))), gmul(a, b)); mattr = matid(3); gcoeff(mattr, 1, 1) = gcopy(c); gcoeff(mattr, 2, 2) = gcopy(alphac); gcoeff(mattr, 3, 3) = gcopy(r); gcoeff(mattr, 2, 3) = gneg(beta); gcoeff(mattr, 1, 2) = gneg(gadd(b, gmul(A, c))); gcoeff(mattr, 1, 3) = gadd(gsub(a, gmul(B, c)), gmul(A, gadd(gmul(A, c), b))); if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" sol de Legendre = "); vec = bnfqfsolve(bnf, alphacp, cp, flag3, auto_s, prec); if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf("%Ps\n", lift(vec)); aux = gmul(gel(vec, 2), dena); gel(vec, 2) = gcopy(gel(vec, 1)); gel(vec, 1) = gcopy(aux); gel(vec, 3) = gmul(gel(vec, 3), denc); vec = gmul(ginv(mattr), vec); vec = gdiv(vec, content(lift(vec))); z1 = gadd(gmul(gadd(gmul(gel(vec, 3), ttheta), gel(vec, 2)), ttheta), gel(vec, 1)); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" z1 = %Ps\n", z1); zc = gmul(zc, gsqr(z1)); if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" zc*z1^2 = %Ps\n", zc); } } } } if (gequal0(sol)) { p24 = cgetg(2, t_VEC); gel(p24, 1) = gcopy(iwhile); listnotELS = concat(listnotELS, p24); iwhile = gaddgs(iwhile, 1); continue; } /* \\\\\\\\\\ */ /* Maintenant zc est de la forme a-b*theta */ /* \\\\\\\\\\ */ if (gequal0(ispointtriv)) { liftzc = lift(zc); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" zc = %Ps\n", liftzc); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" N(zc) = %Ps\n", gnorm(zc)); if (degree(liftzc) >= 2) pari_err(user, " bnfell2descent_gen : c <> 0"); b = gneg(polcoeff0(liftzc, 1, -1)); a = polcoeff0(liftzc, 0, -1); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" on factorise (a,b) = %Ps\n", idealadd(nf, a, b)); ff = idealfactor(nf, idealadd(nf, a, b)); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" ff = %Ps\n", ff); cont = gen_1; l25 = glength(gel(ff, 1)); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; for (i = 1; i <= l25; ++i) { cont = idealmul(nf, cont, idealpow0(nf, gcoeff(ff, i, 1), gdiventgs(gcoeff(ff, i, 2), 2), 0)); if (low_stack(st_lim, stack_lim(btop, 1))) cont = gerepilecopy(btop, cont); } } cont = idealinv(nf, cont); cont = gsqr(basistoalg(nf, gel(bnfisprincipal0(bnf, cont, 3), 2))); a = gmul(a, cont); b = gmul(b, cont); zc = gmul(zc, cont); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) { p26 = cgetg(3, t_VEC); gel(p26, 1) = gcopy(a); gel(p26, 2) = gcopy(b); pari_printf(" [a,b] = %Ps\n", p26); } if (nfissquare(nf, b, prec)) { if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" b est un carre\n"); p27 = cgetg(3, t_VEC); gel(p27, 1) = gdiv(a, b); gel(p27, 2) = gcopy(gel(nfsqrt(nf, gadd(gadd(gadd(gpowgs(gdiv(a, b), 3), gmul(A, gsqr(gdiv(a, b)))), gmul(B, gdiv(a, b))), C), prec), 1)); point = p27; if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf("point trouve = %Ps\n", point); p28 = cgetg(2, t_VEC); gel(p28, 1) = gcopy(point); listpointsmwr = concat(listpointsmwr, p28); m1 = gaddgs(m1, 1); if (gcmp(degre(iwhile), lastloc) > 0) m2 = gaddgs(m2, 1); found = ispointtriv = gen_1; } } /* \\\\\\\\\\\ */ /* Construction de la quartique */ /* \\\\\\\\\\\ */ if (gequal0(ispointtriv)) { r = gcopy(gel(nfsqrt(nf, gnorm(zc), prec), 1)); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" r = %Ps\n", r); c = gmulsg(-2, gadd(gmul(A, b), gmulsg(3, a))); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" c = %Ps\n", c); d = gmulsg(8, r); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" d = %Ps\n", d); e = gsub(gsub(gsub(gmul(gsqr(A), gsqr(b)), gmul(gmul(gmulsg(2, A), a), b)), gmul(gmulsg(4, B), gsqr(b))), gmulsg(3, gsqr(a))); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf(" e = %Ps\n", e); polorig = gmul(gmul(b, gadd(gadd(gadd(gpowgs(x, 4), gmul(c, gsqr(x))), gmul(d, x)), e)), unnf); if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" quartique : (%Ps)*Y^2 = %Ps\n", lift(b), lift(gdiv(polorig, b))); gel(list, 3) = gcopy(b); pol = polorig; if (!gequal0(bigflag)) { redq = bnfredquartique2(bnf, pol, r, a, b); if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" reduite: Y^2 = %Ps\n", lift(gel(redq, 1))); pol = gcopy(gel(redq, 1)); transl = gcopy(gel(redq, 2)); multip = gcopy(gel(redq, 3)); } point = nfratpoint(nf, pol, LIM1, gen_1, prec); if (!gequal(point, cgetg(1, t_VEC))) { if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf("point = %Ps\n", point); m1 = gaddgs(m1, 1); if (!gequal0(bigflag)) { gel(point, 1) = gadd(gmul(gel(point, 1), multip), transl); gel(point, 2) = gcopy(gel(nfsqrt(nf, gsubst(polorig, gvar(x), gdiv(gel(point, 1), gel(point, 3))), prec), 1)); } mattr = matid(3); gcoeff(mattr, 1, 1) = gmulsg(-2, gsqr(b)); gcoeff(mattr, 1, 2) = gmul(gadd(gmul(A, b), a), b); gcoeff(mattr, 1, 3) = gadd(gsqr(a), gmul(gsub(gmulsg(2, B), gsqr(A)), gsqr(b))); gcoeff(mattr, 2, 2) = gneg(b); gcoeff(mattr, 2, 3) = gadd(a, gmul(A, b)); gcoeff(mattr, 3, 3) = gcopy(r); p29 = cgetg(4, t_COL); gel(p29, 1) = gsqr(gel(point, 1)); gel(p29, 2) = gsqr(gel(point, 3)); gel(p29, 3) = gmul(gel(point, 1), gel(point, 3)); UVW = p29; vec = gmul(ginv(mattr), UVW); z1 = gadd(gmul(gadd(gmul(gel(vec, 3), ttheta), gel(vec, 2)), ttheta), gel(vec, 1)); zc = gmul(zc, gsqr(z1)); zc = gdiv(zc, gneg(polcoeff0(lift(zc), 1, -1))); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf("zc*z1^2 = %Ps\n", zc); pointxx = polcoeff0(lift(zc), 0, -1); p30 = cgetg(3, t_VEC); gel(p30, 1) = gcopy(pointxx); gel(p30, 2) = gcopy(gel(nfsqrt(nf, gsubst(gadd(gadd(gadd(gpowgs(x, 3), gmul(A, gsqr(x))), gmul(B, x)), C), gvar(x), pointxx), prec), 1)); point2 = p30; if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) pari_printf(" point trouve = %Ps\n", point2); p31 = cgetg(2, t_VEC); gel(p31, 1) = gcopy(point2); listpointsmwr = concat(listpointsmwr, p31); if (gcmp(degre(iwhile), lastloc) > 0) m2 = gaddgs(m2, 1); found = gen_1; lastloc = gen_m1; } else { if ((!gequal0(loc) || ((gequal0(bigflag)) && nflocallysoluble(nf, pol, r, a, b, prec))) || (!gequal0(bigflag) && nflocallysoluble(nf, pol, gen_0, gen_1, gen_1, prec))) { if (gequal0(loc)) { p32 = cgetg(2, t_VEC); gel(p32, 1) = gcopy(iwhile); listlistELS = concat(listELS, p32); } if (gcmp(degre(iwhile), lastloc) > 0) { m2 = gaddgs(m2, 1); lastloc = degre(iwhile); } point = nfratpoint(nf, pol, LIM3, gen_1, prec); if (!gequal(point, cgetg(1, t_VEC))) { m1 = gaddgs(m1, 1); if (!gequal0(bigflag)) { gel(point, 1) = gadd(gmul(gel(point, 1), multip), transl); gel(point, 2) = gcopy(gel(nfsqrt(nf, gsubst(polorig, gvar(x), gdiv(gel(point, 1), gel(point, 3))), prec), 1)); } mattr = matid(3); gcoeff(mattr, 1, 1) = gmulsg(-2, gsqr(b)); gcoeff(mattr, 1, 2) = gmul(gadd(gmul(A, b), a), b); gcoeff(mattr, 1, 3) = gadd(gsqr(a), gmul(gsub(gmulsg(2, B), gsqr(A)), gsqr(b))); gcoeff(mattr, 2, 2) = gneg(b); gcoeff(mattr, 2, 3) = gadd(a, gmul(A, b)); gcoeff(mattr, 3, 3) = gcopy(r); p33 = cgetg(4, t_COL); gel(p33, 1) = gsqr(gel(point, 1)); gel(p33, 2) = gsqr(gel(point, 3)); gel(p33, 3) = gmul(gel(point, 1), gel(point, 3)); UVW = p33; vec = gmul(ginv(mattr), UVW); z1 = gadd(gmul(gadd(gmul(gel(vec, 3), ttheta), gel(vec, 2)), ttheta), gel(vec, 1)); zc = gmul(zc, gsqr(z1)); zc = gdiv(zc, gneg(polcoeff0(lift(zc), 1, -1))); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf(" zc*z1^2 = %Ps\n", zc); pointxx = polcoeff0(lift(zc), 0, -1); p34 = cgetg(3, t_VEC); gel(p34, 1) = gcopy(pointxx); gel(p34, 2) = gcopy(gel(nfsqrt(nf, gsubst(gadd(gadd(gadd(gpowgs(x, 3), gmul(A, gsqr(x))), gmul(B, x)), C), gvar(x), pointxx), prec), 1)); point2 = p34; if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) pari_printf(" point trouve = %Ps\n", point2); p35 = cgetg(2, t_VEC); gel(p35, 1) = gcopy(point2); listpointsmwr = concat(listpointsmwr, p35); found = gen_1; lastloc = gen_m1; } } else { p36 = cgetg(2, t_VEC); gel(p36, 1) = gcopy(iwhile); listnotELS = concat(listnotELS, p36); } } } if (!gequal0(found)) { found = gen_0; lastloc = gen_m1; v = degre(iwhile); iwhile = shifti(gen_1, gtos(v)); maskwhile = gshift(maskwhile, -1); LS2coordtilda = extract0(LS2coordtilda, gsubgs(gsub(shifti(gen_1, glength(listgen)), iwhile), 1), NULL); listgen = extract0(listgen, gsubgs(gsub(shifti(gen_1, glength(listgen)), iwhile), 1), NULL); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); while (gcmp(gel(listELS, glength(listELS)), iwhile) >= 0) { listELS = extract0(listELS, subis(shifti(gen_1, glength(listELS) - 1), 1), NULL); if (low_stack(st_lim, stack_lim(btop, 1))) listELS = gerepilecopy(btop, listELS); } } { pari_sp btop = avma, st_lim = stack_lim(btop, 1); while (glength(listnotELS) && (gcmp(gel(listnotELS, glength(listnotELS)), iwhile) >= 0)) { listnotELS = extract0(listnotELS, subis(shifti(gen_1, glength(listnotELS) - 1), 1), NULL); if (low_stack(st_lim, stack_lim(btop, 1))) listnotELS = gerepilecopy(btop, listnotELS); } } } else iwhile = gaddgs(iwhile, 1); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 61, &sol, &loc, &listnotELS, &iwhile, &listELS, &p20, &iaux, &p22, &zc, &liftzc, &ispointtriv, &point, &listpointsmwr, &m1, &m2, &found, &c, &b, &a, &alphac, &r, &denc, &cp, &dena, &alphacp, &beta, &mattr, &vec, &aux, &z1, &p24, &ff, &cont, &p26, &p27, &p28, &d, &e, &polorig, &list, &pol, &redq, &transl, &multip, &p29, &UVW, &pointxx, &p30, &point2, &p31, &lastloc, &p32, &listlistELS, &p33, &p34, &p35, &p36, &v, &maskwhile, &LS2coordtilda, &listgen); } } if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) { pari_printf("m1 = %Ps\n", m1); pari_printf("m2 = %Ps\n", m2); } if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) pari_printf("#S(E/K)[2] = %Ps\n", shifti(gen_1, gtos(m2))); if (gequal(m1, m2)) { if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) { pari_printf("#E(K)/2E(K) = %Ps\n", shifti(gen_1, gtos(m1))); pari_printf("#III(E/K)[2] = 1\n"); pari_printf("rang(E/K) = %Ps\n", m1); } rang = m1; } else { if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) { pari_printf("#E(K)/2E(K) >= %Ps\n", shifti(gen_1, gtos(m1))); pari_printf("#III(E/K)[2] <= %Ps\n", shifti(gen_1, gtos(gsub(m2, m1)))); pari_printf("rang(E/K) >= %Ps\n", m1); } rang = m1; if (!gequal0(gmodgs(gsub(m2, m1), 2))) { if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) { pari_printf(" III devrait etre un carre, donc \n"); if (gcmpgs(gsub(m2, m1), 1) > 0) { pari_printf("#E(K)/2E(K) >= %Ps\n", shifti(gen_1, gtos(gaddgs(m1, 1)))); pari_printf("#III(E/K)[2] <= %Ps\n", shifti(gen_1, gtos(gsubgs(gsub(m2, m1), 1)))); pari_printf("rang(E/K) >= %Ps\n", gaddgs(m1, 1)); } else { pari_printf("#E(K)/2E(K) = %Ps\n", shifti(gen_1, gtos(gaddgs(m1, 1)))); pari_printf("#III(E/K)[2] = 1\n"); pari_printf("rang(E/K) = %Ps\n", gaddgs(m1, 1)); } } rang = gaddgs(m1, 1); } } if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) pari_printf("listpointsmwr = %Ps\n", listpointsmwr); l14 = glength(listpointsmwr); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; for (i = 1; i <= l14; ++i) { if (glength(gel(listpointsmwr, i)) == 3) gel(listpointsmwr, i) = extract0(gel(listpointsmwr, i), stoi(3), NULL); if (gequal0(ellisoncurve(ellnf, gel(listpointsmwr, i)))) pari_err(user, "bnfell2descent : MAUVAIS POINT "); if (low_stack(st_lim, stack_lim(btop, 1))) listpointsmwr = gerepilecopy(btop, listpointsmwr); } } if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("fin de bnfell2descent_gen\n"); p15 = cgetg(4, t_VEC); gel(p15, 1) = gcopy(rang); gel(p15, 2) = gcopy(m2); gel(p15, 3) = gcopy(listpointsmwr); p15 = gerepilecopy(ltop, p15); return p15; } GEN bnfellrank(GEN bnf, GEN ell, GEN help, GEN bigflag, GEN flag3, long prec) { pari_sp ltop = avma; GEN urst = gen_0, urst1 = gen_0, den = gen_0, factden = gen_0, eqtheta = gen_0, rnfeq = gen_0, bbnf = gen_0, ext = gen_0, rang = gen_0, f = gen_0; GEN p1 = gen_0, p2 = gen_0, p3 = gen_0, p4 = gen_0; /* vec */ GEN y = pol_x(fetch_user_var("y")), x = pol_x(fetch_user_var("x")); GEN p5 = gen_0, p6 = gen_0; /* vec */ if (!help) help = cgetg(1, t_VEC); if (!bigflag) bigflag = gen_1; if (!flag3) flag3 = gen_1; /* MODI add f */ if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf("entree dans bnfellrank\n"); if (glength(ell) <= 5) ell = smallellinit(ell); p1 = cgetg(5, t_VEC); gel(p1, 1) = gen_1; gel(p1, 2) = gen_0; gel(p1, 3) = gen_0; gel(p1, 4) = gen_0; /* removes the coefficients a1 and a3 */ urst = p1; if (!gequalgs(ell_get_a1(ell), 0) || !gequalgs(ell_get_a3(ell), 0)) { p2 = cgetg(5, t_VEC); gel(p2, 1) = gen_1; gel(p2, 2) = gen_0; gel(p2, 3) = gdivgs(gneg(ell_get_a1(ell)), 2); gel(p2, 4) = gdivgs(gneg(ell_get_a3(ell)), 2); urst1 = p2; ell = ellchangecurve(ell, urst1); urst = ellcomposeurst(urst, urst1); } /* removes denominators */ { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long l7; GEN p8 = gen_0; /* vec */ while (gcmpgs(gcoeff(den = idealinv(bnf, idealadd(bnf, idealadd(bnf, gen_1, ell_get_a2(ell)), idealadd(bnf, ell_get_a4(ell), ell_get_a6(ell)))), 1, 1), 1) > 0) { factden = gcopy(gel(idealfactor(bnf, den), 1)); den = gen_1; l7 = glength(factden); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; for (i = 1; i <= l7; ++i) { den = idealmul(bnf, den, gel(factden, i)); if (low_stack(st_lim, stack_lim(btop, 1))) den = gerepilecopy(btop, den); } } den = gcopy(gcoeff(den, 1, 1)); p8 = cgetg(5, t_VEC); gel(p8, 1) = ginv(den); gel(p8, 2) = gen_0; gel(p8, 3) = gen_0; gel(p8, 4) = gen_0; urst1 = p8; ell = ellchangecurve(ell, urst1); urst = ellcomposeurst(urst, urst1); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 6, &den, &factden, &p8, &urst1, &ell, &urst); } } help = ellchangepoint(help, urst); /* choix de l'algorithme suivant la 2-torsion */ ell = gmul(/* choix de l'algorithme suivant la 2-torsion */ ell, gmodulsg(1, member_pol(bnf))); p3 = cgetg(5, t_VEC); gel(p3, 1) = gen_1; gel(p3, 2) = gcopy(ell_get_a2(ell)); gel(p3, 3) = gcopy(ell_get_a4(ell)); gel(p3, 4) = gcopy(ell_get_a6(ell)); eqtheta = gtopoly(p3, -1); if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) pari_printf("courbe elliptique : Y^2 = %Ps\n", eqtheta); f = nfpolratroots(bnf, eqtheta); if (glength(f) == 0) { /* cas 1: 2-torsion triviale */ rnfeq = rnfequation0(bnf, eqtheta, 1); p4 = cgetg(5, t_VEC); gel(p4, 1) = gen_1; gel(p4, 2) = gmul(gneg(gel(rnfeq, 3)), gmodulo(y, member_pol(bnf))); gel(p4, 3) = gen_0; gel(p4, 4) = gen_0; urst1 = p4; if (!gequalgs(gel(rnfeq, 3), 0)) { ell = ellchangecurve(ell, urst1); urst = ellcomposeurst(urst, urst1); eqtheta = gsubst(eqtheta, gvar(x), gsub(x, gmul(gel(rnfeq, 3), gmodulo(y, member_pol(bnf))))); rnfeq = rnfequation0(bnf, eqtheta, 1); if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf("translation : on travaille avec Y^2 = %Ps\n", eqtheta); } if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf("bbnfinit "); bbnf = Buchall(gel(rnfeq, 1), nf_FORCE, prec); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf("done\n"); p5 = cgetg(4, t_VEC); gel(p5, 1) = gcopy(eqtheta); gel(p5, 2) = gcopy(rnfeq); gel(p5, 3) = gcopy(bbnf); ext = p5; rang = bnfell2descent_gen(bnf, ell, ext, help, bigflag, flag3, NULL, prec); } else { if (glength(f) == 1) { /* cas 2: 2-torsion = Z/2Z */ if (!gequalgs(gel(f, 1), 0)) { p6 = cgetg(5, t_VEC); gel(p6, 1) = gen_1; gel(p6, 2) = gcopy(gel(f, 1)); gel(p6, 3) = gen_0; gel(p6, 4) = gen_0; urst1 = p6; ell = ellchangecurve(ell, urst1); urst = ellcomposeurst(urst, urst1); } rang = bnfell2descent_viaisog(bnf, ell, prec); } else /* cas 3: 2-torsion = Z/2Z*Z/2Z */ rang = bnfell2descent_complete(bnf, gel(f, 1), gel(f, 2), gel(f, 3), flag3, NULL, prec); } gel(rang, 3) = ellchangepointinverse(gel(rang, 3), urst); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf("fin de bnfellrank\n"); rang = gerepilecopy(ltop, rang); return rang; } GEN bnfell2descent_complete(GEN bnf, GEN e1, GEN e2, GEN e3, GEN flag3, GEN auto_s, long prec) /* vec */ { pari_sp ltop = avma; GEN p1 = gen_0; /* vec */ GEN y = pol_x(fetch_user_var("y")), KS2prod = gen_0, oddclass = gen_0, KS2gen = gen_0, vect = gen_0, selmer = gen_0, rang = gen_0, b1 = gen_0, b2 = gen_0, vec = gen_0, z1 = gen_0, z2 = gen_0, d31 = gen_0, quart0 = gen_0, quart = gen_0, cont = gen_0, fa = gen_0, point = gen_0, solx = gen_0, soly = gen_0, listepoints = gen_0, strange = gen_0; long l2, l3; GEN p4 = gen_0; /* vec */ GEN x = pol_x(fetch_user_var("x")); GEN p5 = gen_0; /* vec */ if (!flag3) flag3 = gen_1; if (!auto_s) { p1 = cgetg(2, t_VEC); gel(p1, 1) = gcopy(y); auto_s = p1; } /* MODI remove i,X,Y */ /* MODI add strange */ if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf("Algorithme de la 2-descente complete\n"); /* calcul de K(S,2) */ KS2prod = gmulgs(gmul(gmul(gsub(e1, e2), gsub(e2, e3)), gsub(e3, e1)), 2); oddclass = gen_0; { pari_sp btop = avma, st_lim = stack_lim(btop, 1); while (gequal0(oddclass)) { KS2gen = bnfsunit(bnf, gtrans(gel(idealfactor(bnf, KS2prod), 1)), prec); oddclass = gmodgs(gel(gel(KS2gen, 5), 1), 2); if (gequal0(oddclass)) KS2prod = idealmul(bnf, KS2prod, gel(gel(gel(KS2gen, 5), 3), 1)); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 3, &KS2gen, &oddclass, &KS2prod); } } KS2gen = gcopy(gel(KS2gen, 1)); /* A CHANGER : KS2gen = matbasistoalg(bnf,KS2gen); */ l2 = glength(KS2gen); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; for (i = 1; i <= l2; ++i) { gel(KS2gen, i) = basistoalg(bnf, gel(KS2gen, i)); if (low_stack(st_lim, stack_lim(btop, 1))) KS2gen = gerepilecopy(btop, KS2gen); } } KS2gen = concat(gmodulo(lift(gettufu(bnf)), member_pol(bnf)), KS2gen); if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) { pari_printf("#K(S,2)gen = %ld\n", glength(KS2gen)); pari_printf("K(S,2)gen = %Ps\n", KS2gen); } l3 = glength(KS2gen); { long i; GEN p6 = gen_0; /* vec */ p4 = cgetg(l3+1, t_VEC); for (i = 1; i <= l3; ++i) { p6 = cgetg(3, t_VEC); gel(p6, 1) = gen_0; gel(p6, 2) = gen_1; gel(p4, i) = p6; } } /* parcours de K(S,2)*K(S,2) */ vect = p4; selmer = gen_0; rang = gen_0; listepoints = cgetg(1, t_VEC); { GEN fv_data = gen_0; GEN (*fv_next)(GEN, GEN); /* func_GG */ GEN X = forvec_start(vect, 0, &fv_data, &fv_next); /* vec */ { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long l7; GEN p8 = gen_0; for ( ; X; X = fv_next(fv_data, X)) { l7 = glength(KS2gen); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; p8 = gen_1; for (i = 1; i <= l7; ++i) { p8 = gmul(p8, gpow(gel(KS2gen, i), gel(X, i), prec)); if (low_stack(st_lim, stack_lim(btop, 1))) p8 = gerepilecopy(btop, p8); } } b1 = p8; { GEN fv_data = gen_0; GEN (*fv_next)(GEN, GEN); /* func_GG */ GEN Y = forvec_start(vect, 0, &fv_data, &fv_next); /* vec */ { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long l9; GEN p10 = gen_0; GEN p11 = gen_0, p12 = gen_0; /* vec */ long l13; GEN p14 = gen_0, p15 = gen_0, p16 = gen_0; /* vec */ for ( ; Y; Y = fv_next(fv_data, Y)) { l9 = glength(KS2gen); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; p10 = gen_1; for (i = 1; i <= l9; ++i) { p10 = gmul(p10, gpow(gel(KS2gen, i), gel(Y, i), prec)); if (low_stack(st_lim, stack_lim(btop, 1))) p10 = gerepilecopy(btop, p10); } } b2 = p10; if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) { p11 = cgetg(3, t_VEC); gel(p11, 1) = gcopy(b1); gel(p11, 2) = gcopy(b2); pari_printf("[b1,b2] = %Ps\n", lift(p11)); } /* points triviaux provenant de la 2-torsion */ if (gequal1(b1) && gequal1(b2)) { if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" point trivial [0]\n"); selmer = gaddgs(selmer, 1); rang = gaddgs(rang, 1); continue; } if (nfissquare(member_nf(bnf), gmul(gsub(e2, e1), b1), prec) && nfissquare(member_nf(bnf), gmul(gmul(gsub(e2, e3), gsub(e2, e1)), b2), prec)) { if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" point trivial [e2,0]\n"); selmer = gaddgs(selmer, 1); rang = gaddgs(rang, 1); continue; } if (nfissquare(member_nf(bnf), gmul(gsub(e1, e2), b2), prec) && nfissquare(member_nf(bnf), gmul(gmul(gsub(e1, e3), gsub(e1, e2)), b1), prec)) { if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" point trivial [e1,0]\n"); selmer = gaddgs(selmer, 1); rang = gaddgs(rang, 1); continue; } if (nfissquare(member_nf(bnf), gmul(gsub(e3, e1), b1), prec) && nfissquare(member_nf(bnf), gmul(gsub(e3, e2), b2), prec)) { if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" point trivial [e3,0]\n"); selmer = gaddgs(selmer, 1); rang = gaddgs(rang, 1); continue; } /* premier critere local : sur les formes quadratiques */ if (((mynfhilbert(member_nf(bnf), gmul(b1, b2), gmul(b1, gsub(e2, e1)), prec) < 0) || (mynfhilbert(member_nf(bnf), b2, gmul(b1, gsub(e3, e1)), prec) < 0)) || (mynfhilbert(member_nf(bnf), b1, gmul(b2, gsub(e3, e2)), prec) < 0)) { if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf("non ELS\n"); continue; } if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) { p12 = cgetg(3, t_VEC); gel(p12, 1) = gcopy(b1); gel(p12, 2) = gcopy(b2); pari_printf("[b1,b2] = %Ps\n", lift(p12)); } if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf("qf loc soluble\n"); /* solution de la premiere forme quadratique */ if (!gequal(b1, b2)) { vec = bnfqfsolve(bnf, gmul(b1, b2), gmul(b1, gsub(e2, e1)), flag3, NULL, prec); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf("sol part = %Ps\n", vec); if (gequal0(gel(vec, 3))) pari_err(user, "bnfell2descent_complete : BUG !!! : vec[3]=0 "); z1 = gdiv(gdiv(gel(vec, 1), gel(vec, 3)), b1); z2 = gdiv(gel(vec, 2), gel(vec, 3)); } else { z1 = gdivgs(gaddsg(1, gdiv(gsub(e2, e1), b1)), 2); z2 = gsubgs(z1, 1); } d31 = gsub(e3, e1); quart0 = gadd(gsub(gadd(gsub(gmul(gmul(gsqr(b2), gsub(gmul(gsqr(z1), b1), d31)), gpowgs(x, 4)), gmul(gmul(gmul(gmul(gmulsg(4, z1), gsqr(b2)), z2), b1), gpowgs(x, 3))), gmul(gmul(gmul(gmulsg(2, b1), b2), gadd(gadd(gmul(gsqr(z1), b1), gmul(gmulsg(2, b2), gsqr(z2))), d31)), gsqr(x))), gmul(gmul(gmul(gmul(gmulsg(4, z1), b2), z2), gsqr(b1)), x)), gmul(gsqr(b1), gsub(gmul(gsqr(z1), b1), d31))); quart = gmul(gmul(quart0, b1), b2); if (gcmpgs(DEBUGLEVEL_ell, 4) >= 0) pari_printf("quart = %Ps\n", quart); quart = gmul(quart, gsqr(denom(simplify(content(quart))))); cont = simplify(content(lift(quart))); fa = factor(cont); l13 = glength(gel(fa, 1)); { pari_sp btop = avma, st_lim = stack_lim(btop, 1); long i; for (i = 1; i <= l13; ++i) { quart = gdiv(quart, gpow(gcoeff(fa, i, 1), gmulsg(2, gdiventgs(gcoeff(fa, i, 2), 2)), prec)); if (low_stack(st_lim, stack_lim(btop, 1))) quart = gerepilecopy(btop, quart); } } if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf("quart red = %Ps\n", quart); /* la quartique est-elle localement soluble ? */ if (!nflocallysoluble(member_nf(bnf), quart, NULL, NULL, NULL, prec)) { if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf(" quartique non ELS \n"); continue; } selmer = gaddgs(selmer, 1); /* recherche de points sur la quartique. */ point = nfratpoint(member_nf(bnf), quart, LIM3, gen_1, prec); if (!gequal(point, cgetg(1, t_VEC))) { if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf("point trouve sur la quartique !!\n"); if (gcmpgs(DEBUGLEVEL_ell, 3) >= 0) pari_printf("%Ps\n", point); if (!gequal0(gel(point, 3))) { point = gdiv(point, gel(point, 3)); z1 = gdiv(gsub(gmul(gmul(gmulsg(2, b2), gel(point, 1)), z2), gmul(z1, gadd(b1, gmul(b2, gsqr(gel(point, 1)))))), gsub(b1, gmul(b2, gsqr(gel(point, 1))))); solx = gadd(gmul(b1, gsqr(z1)), e1); soly = gcopy(gel(nfsqrt(member_nf(bnf), gmul(gmul(gsub(solx, e1), gsub(solx, e2)), gsub(solx, e3)), prec), 1)); p14 = cgetg(2, t_VEC); p15 = cgetg(3, t_VEC); gel(p15, 1) = gcopy(solx); gel(p15, 2) = gcopy(soly); gel(p14, 1) = p15; listepoints = concat(listepoints, p14); if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) { p16 = cgetg(3, t_VEC); gel(p16, 1) = gcopy(solx); gel(p16, 2) = gcopy(soly); pari_printf("point sur la courbe elliptique =%Ps\n", p16); } } rang = gaddgs(rang, 1); } else if (gcmpgs(DEBUGLEVEL_ell, 2) >= 0) pari_printf("aucun point trouve sur la quartique\n"); if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 22, &p10, &b2, &p11, &selmer, &rang, &p12, &vec, &z1, &z2, &d31, &quart0, &quart, &cont, &fa, &point, &solx, &soly, &p15, &p14, &listepoints, &p16, &Y); } } } if (low_stack(st_lim, stack_lim(btop, 1))) gerepileall(btop, 18, &p8, &b1, &b2, &selmer, &rang, &vec, &z1, &z2, &d31, &quart0, &quart, &cont, &fa, &point, &solx, &soly, &listepoints, &X); } } } /* fin */ if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) pari_printf("#S^(2) = %Ps\n", selmer); if (gcmp(rang, gdivgs(selmer, 2)) > 0) rang = selmer; if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) { strange = stoi(!gequal(rang, selmer)); if (!gequal0(strange)) pari_printf("#E[K]/2E[K]>= %Ps\n", rang); else pari_printf("#E[K]/2E[K] = %Ps\n", rang); pari_printf("#E[2] = 4\n"); } rang = gsubgs(gceil(gdiv(glog(rang, prec), glog(gen_2, prec))), 2); selmer = stoi(ggval(selmer, gen_2)); if (gcmpgs(DEBUGLEVEL_ell, 1) >= 0) { if (!gequal0(strange)) pari_printf("%Ps >= rang >= %Ps\n", gsubgs(selmer, 2), rang); else pari_printf("rang = %Ps\n", rang); if (!gequal0(rang)) pari_printf("points = %Ps\n", listepoints); } p5 = cgetg(4, t_VEC); gel(p5, 1) = gcopy(rang); gel(p5, 2) = gcopy(selmer); gel(p5, 3) = gcopy(listepoints); p5 = gerepilecopy(ltop, p5); return p5; }