was@form:~/talks/magma_ihp/bsd$ was@form:~/talks/magma_ihp/bsd$ Magma V2.11-8 Wed Oct 13 2004 13:18:57 on form [Seed = 3656776467]
Type ? for help. Type -D to quit.
Loading startup file "/home/was/magma/local/emacs.m"
Loading "/home/was/magma/local/init.m"
> > >
>
>
> J := J0(37);
> J;
Modular abelian variety JZero(37) of dimension 2 and level 37
over Q
> Decomposition(J);
[
Modular abelian variety 37A of dimension 1, level 37 and
conductor 37 over Q,
Modular abelian variety 37B of dimension 1, level 37 and
conductor 37 over Q
]
> S := CuspForms(37);
> N := Newforms(S);
> N;
[* [*
q - 2*q^2 - 3*q^3 + 2*q^4 - 2*q^5 + 6*q^6 - q^7 + O(q^8)
*], [*
q + q^3 - 2*q^4 - q^7 + O(q^8)
*] *]
> f := N[1][1];
> Af := ModularAbelianVariety(f);
> Af;
Modular abelian variety Af of dimension 1 and level 37 over Q
> J := J0(389);
> Decomposition(J);
[
Modular abelian variety 389A of dimension 1, level 389 and
conductor 389 over Q,
Modular abelian variety 389B of dimension 2, level 389 and
conductor 389^2 over Q,
Modular abelian variety 389C of dimension 3, level 389 and
conductor 389^3 over Q,
Modular abelian variety 389D of dimension 6, level 389 and
conductor 389^6 over Q,
Modular abelian variety 389E of dimension 20, level 389 and
conductor 389^20 over Q
]
> D := $1;
> EllipticCurve(D[1]);
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational
Field
> EllipticCurve(D[2]);
>> EllipticCurve(D[2]);
^
Runtime error in 'EllipticCurve': Argument 1 must have dimension
1.
> D[2];
Modular abelian variety 389B of dimension 2, level 389 and
conductor 389^2 over Q
> E := D[1];
> E;
Modular abelian variety 389A of dimension 1, level 389 and
conductor 389 over Q
> L := LSeries(E);
> L;
L(389A,s): L-series of Modular abelian variety 389A of dimension
1, level 389 and conductor 389 over Q
> alpha, r := LeadingCoefficient(L,1, 300);
> alpha;
0.75931650029224679065762600319
> r;
2
> EE := EllipticCurve(E);
> r, alpha := LeadingCoefficient(L,1, 300);
> r, alpha := AnalyticRank(EE);
> r;
2
> alpha;
0.7593000000
> J := J0(125);
> D := Decomposition(J);
> D;
[
Modular abelian variety 125A of dimension 2, level 5^3 and
conductor 5^6 over Q,
Modular abelian variety 125B of dimension 2, level 5^3 and
conductor 5^6 over Q,
Modular abelian variety 125C of dimension 4, level 5^3 and
conductor 5^12 over Q
]
> TorsionMultiple(D[3], 2);
0
> TorsionMultiple(D[3], 3);
155
> TorsionMultiple(D[3], 13);
5
> TorsionMultiple(D[3], 37);
5
> [ : A in D];
[ <2, 1>, <2, 5>, <4, 5> ]
> RationalCuspidalSubgroup(D[1]);
Finitely generated subgroup of abelian variety with invariants []
> RationalCuspidalSubgroup(D[2]);
Finitely generated subgroup of abelian variety with invariants [
5 ]
> RationalCuspidalSubgroup(D[3]);
Finitely generated subgroup of abelian variety with invariants [
5 ]
> RationalCuspidalSubgroup(J);
Finitely generated subgroup of abelian variety with invariants [
25 ]
> D[2] meet D[3];
Finitely generated subgroup of abelian variety with invariants [
5, 5, 5, 5 ]
Modular abelian variety ZERO of dimension 0 and level 5^3 over Q
Homomorphism from modular abelian variety of dimension 0 to 125B
x 125C given on integral homology by:
Matrix with 0 rows and 12 columns
Homomorphism from 125B x 125C to JZero(125) (not printing 12x16
matrix)
> A := D[2]/RationalCuspidalSubgroup(D[2]);
> A;
Modular abelian variety of dimension 2 and level 5^3 over Q
> IsIsomorphic(A,D[2]);
false
>