JMM2012 -- solutions
system:sage


{{{id=1|
K.<d> = QQ[]
E = EllipticCurve([1+d-d^2, d^2-d^3, d^2-d^3, 0, 0])
E
///
Elliptic Curve defined by y^2 + (-d^2+d+1)*x*y + (-d^3+d^2)*y = x^3 + (-d^3+d^2)*x^2 over Univariate Polynomial Ring in d over Rational Field
}}}

{{{id=2|
E.short_weierstrass_model()
///
Elliptic Curve defined by y^2 = x^3 + (-27*d^8+324*d^7-1134*d^6+1512*d^5-945*d^4+378*d^2-108*d-27)*x + (54*d^12-972*d^11+6318*d^10-19116*d^9+30780*d^8-26244*d^7+14742*d^6-11988*d^5+9396*d^4-2484*d^3-810*d^2+324*d+54) over Univariate Polynomial Ring in d over Rational Field
}}}

{{{id=3|
show(E.short_weierstrass_model())
///
<html><div class="math">\newcommand{\Bold}[1]{\mathbf{#1}}y^2 = x^3 + \left(-27 d^{8} + 324 d^{7} - 1134 d^{6} + 1512 d^{5} - 945 d^{4} + 378 d^{2} - 108 d - 27\right)x + \left(54 d^{12} - 972 d^{11} + 6318 d^{10} - 19116 d^{9} + 30780 d^{8} - 26244 d^{7} + 14742 d^{6} - 11988 d^{5} + 9396 d^{4} - 2484 d^{3} - 810 d^{2} + 324 d + 54\right) </div></html>
}}}

{{{id=5|

///
}}}

{{{id=4|
for d in [1..1000]:
    if is_fundamental_discriminant(d):
        K = QuadraticField(d)
        if K.class_number() == 5:
            print "d = ", d
            break
///
d =  401
}}}

{{{id=6|
for d in [-1,-2,..,-1000]:
    if is_fundamental_discriminant(d):
        K = QuadraticField(d)
        if K.class_number() == 5:
            print "d = ", d
            break
///
d =  -47
}}}

{{{id=8|
R.<x> = QQ[]
for d in [1..500]:
    f = x^3 + d
    if not f.is_irreducible(): continue
    K = NumberField(f, 'a')
    if K.class_number() == 5:
        print K
        break
///
Number Field in a with defining polynomial x^3 + 263
}}}

{{{id=7|

///
}}}

{{{id=10|
def E(a): 
    return EllipticCurve([0,(a-1),1,-a,0])

for a in [0..80]:
    print a, E(a).rank()
///
0 0
1 1
2 2
3 2
4 3
5 2
6 2
7 3
8 3
9 3
10 2
11 3
12 3
13 3
14 3
15 2
16 4
17 3
18 2
19 3
20 3
21 2
22 3
23 4
24 3
25 3
26 3
27 2
28 3
29 3
30 2
31 3
32 3
33 2
34 4
35 2
36 3
37 4
38 3
39 2
40 3
41 2
42 4
43 3
44 2
45 2
46 4
47 3
48 3
49 4
50 2
51 2
52 2
53 4
54 4
55 2
56 3
57 3
58 3
59 4
60 3
61 2
62 3
63 3
64 4
65 3
66 3
67 4
68 2
69 3
70 3
71 3
72 3
73 3
74 4
75 3
76 3
77 3
78 3
79 5
80 3
}}}

{{{id=11|
# so smallest are: 0,1,2,4,16,79.
///
}}}

{{{id=12|

///
}}}