SD14: Elliptic curves
system:sage

{{{id=0|
E = EllipticCurve([-1,0]); E
///
Elliptic Curve defined by y^2 = x^3 - x over Rational Field
}}}

{{{id=1|
# sanity check
E.genus()
///
1
}}}

{{{id=2|
E.discriminant()
///
64
}}}

{{{id=3|
E.conductor()
///
32
}}}

{{{id=4|
E/GF(next_prime(1000000))
///
Elliptic Curve defined by y^2 = x^3 + 1000002*x over Finite Field of size 1000003
}}}

{{{id=5|
E/GF(2)
///
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "/home/aghitza/.sage/sage_notebook/worksheets/admin/5/code/7.py", line 7, in <module>
    exec compile(ur'E/GF(_sage_const_2 )' + '\n', '', 'single')
  File "/opt/sage/local/lib/python2.5/site-packages/SQLAlchemy-0.4.6-py2.5.egg/", line 1, in <module>
    
  File "/opt/sage/local/lib/python2.5/site-packages/sage/schemes/generic/scheme.py", line 253, in __div__
    return self.base_extend(Y)
  File "/opt/sage/local/lib/python2.5/site-packages/sage/schemes/elliptic_curves/ell_generic.py", line 1029, in base_extend
    return constructor.EllipticCurve([R(a) for a in self.a_invariants()])
  File "/opt/sage/local/lib/python2.5/site-packages/sage/schemes/elliptic_curves/constructor.py", line 223, in EllipticCurve
    return ell_finite_field.EllipticCurve_finite_field(x, y)
  File "/opt/sage/local/lib/python2.5/site-packages/sage/schemes/elliptic_curves/ell_finite_field.py", line 75, in __init__
    EllipticCurve_field.__init__(self, ainvs)
  File "/opt/sage/local/lib/python2.5/site-packages/sage/schemes/elliptic_curves/ell_generic.py", line 147, in __init__
    "Invariants %s define a singular curve."%ainvs
ArithmeticError: Invariants [0, 0, 0, 1, 0] define a singular curve.
}}}

{{{id=11|
E.has_additive_reduction(2)
///
True
}}}

{{{id=17|
E.supersingular_primes(100)
///
[3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83]
}}}

{{{id=6|
(E/GF(3)).automorphisms()
///
[Generic endomorphism of Abelian group of points on Elliptic Curve defined by y^2 = x^3 + 2*x over Finite Field of size 3
  Via:  (u,r,s,t) = (1, 0, 0, 0), Generic endomorphism of Abelian group of points on Elliptic Curve defined by y^2 = x^3 + 2*x over Finite Field of size 3
  Via:  (u,r,s,t) = (1, 1, 0, 0), Generic endomorphism of Abelian group of points on Elliptic Curve defined by y^2 = x^3 + 2*x over Finite Field of size 3
  Via:  (u,r,s,t) = (1, 2, 0, 0), Generic endomorphism of Abelian group of points on Elliptic Curve defined by y^2 = x^3 + 2*x over Finite Field of size 3
  Via:  (u,r,s,t) = (2, 0, 0, 0), Generic endomorphism of Abelian group of points on Elliptic Curve defined by y^2 = x^3 + 2*x over Finite Field of size 3
  Via:  (u,r,s,t) = (2, 1, 0, 0), Generic endomorphism of Abelian group of points on Elliptic Curve defined by y^2 = x^3 + 2*x over Finite Field of size 3
  Via:  (u,r,s,t) = (2, 2, 0, 0)]
}}}

{{{id=8|
# number of points of reduction mod p
time E.Np(next_prime(10^100))
///
10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000268
Time: CPU 0.30 s, Wall: 0.40 s
}}}

{{{id=10|
E.multiplication_by_m(2)
///
((x^4 + 2*x^2 + 1)/(4*x^3 - 4*x), (8*x^6*y - 40*x^4*y - 40*x^2*y + 8*y)/(64*x^6 - 128*x^4 + 64*x^2))
}}}

{{{id=12|
E.rank()
///
0
}}}

{{{id=13|
E.integral_points()
///
[(-1 : 0 : 1), (0 : 0 : 1), (1 : 0 : 1)]
}}}

{{{id=14|
# generators of the torsion-free part of the Mordell-Weil group
E.gens()
///
[]
}}}

{{{id=16|
E.plot()
///
}}}

{{{id=18|
f = E.modular_form(); f
///
q - 2*q^5 + O(q^6)
}}}

{{{id=19|
f.q_expansion(100)
///
q - 2*q^5 - 3*q^9 + 6*q^13 + 2*q^17 - q^25 - 10*q^29 - 2*q^37 + 10*q^41 + 6*q^45 - 7*q^49 + 14*q^53 - 10*q^61 - 12*q^65 - 6*q^73 + 9*q^81 - 4*q^85 + 10*q^89 + 18*q^97 + O(q^100)
}}}

{{{id=21|
ls = E.lseries(); ls
///
Complex L-series of the Elliptic Curve defined by y^2 = x^3 - x over Rational Field
}}}

{{{id=22|
ls.taylor_series(series_prec=5)
///
0.655514388573030 + 0.447208159472739*z - 0.233131198781643*z^2 + 0.0342258563577268*z^3 + 0.0291414074964433*z^4 + O(z^5)
}}}

{{{id=24|
ls(1)
///
0.655514388573030
}}}

{{{id=25|
# first 10 zeros on the critical line
ls.zeros(10)
///
[3.67478223, 5.87146419, 7.77199474, 8.95538623, 10.9076921, 11.7666127, 13.2768755, 14.5765256, 15.7488207, 16.7385637]
}}}

{{{id=26|
# label in Cremona's database
E.label()
///
'32a2'
}}}

{{{id=27|
E = EllipticCurve('2001a'); E
///
Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 + 11*x - 52 over Rational Field
}}}

{{{id=29|

///
}}}