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Modular abelian varieties of dimension 1 are elliptic curves.
Given a modular abelian variety A over Q of dimension 1,
Magma can compute an elliptic curve that is isogenous
over Q to A. Given an elliptic curve E over Q,
a modular abelian variety over Q
that is isogenous to E can be constructed.
It would be very desirable to make these commands more precise, and to
extend them to work over other fields. For example, modular abelian
varieties should (conjecturally) be associated to Q-curves and
their restriction of scalars.
An elliptic curve isogenous to the modular abelian variety A over
the rational field, if there is an elliptic curve associated to A. For
A of weight greater than 2 use the EllipticInvariants command.
Sign: RngIntElt Default: 0
A modular abelian variety isogenous to the elliptic curve E.
Note that elliptic
curves with small coefficients can have quite large conductor, hence
computing the massive modular abelian variety that has E as quotient, which
is one thing this function does, could take some time.
We apply the above two commands to the elliptic curve J 0(49).
> A := JZero(49);
> E := EllipticCurve(A); E;
Elliptic Curve defined by y^2 + x*y = x^3 - x^2 - 2*x - 1 over
Rational Field
> B := ModularAbelianVariety(E); B;
Modular abelian variety 'Cremona 49A' of dimension 1 and level
7^2 over Q
To see how A and B compare, we first test equality and
see they are not equal (since they were constructed differently).
However, they are isomorphic.
> B eq A;
false
> IsIsomorphic(A,B);
true Homomorphism from JZero(49) to 'Cremona 49A' given on integral
homology by:
[1 0]
[0 1]
> phi := NaturalMap(A,B);
> Degree(phi);
1
> phi;
Homomorphism N(1) from JZero(49) to 'Cremona 49A' given on integral
homology by:
[1 0]
[0 1]
Thus B is embedded in A via the identity map.
Let A be an abelian variety over Q of dimension 1. The
following two functions use standard iterative algorithms (see
Cremona's book) to compute the invariants c4, c6, j, and
generators of the period lattice of the optimal quotient of J0(N)
associated to A.
Invariants c4, c6, j, and an elliptic curve, of the one
dimensional modular abelian variety A, computed using n terms of
q-expansion.
Elliptic periods w1 and w2 of the J0(N)-optimal elliptic
curve associated to the modular abelian variety
A, computed using n terms of the q-expansion. The
periods have the property that w1/w2 has positive imaginary part.
> A := ModularAbelianVariety("100A");
> c4,c6,j,E := EllipticInvariants(A,100);
> c4;
1600.0523183040458033068678491117208 + 0.E-25*i
> c6;
-44002.166592330033618811790218678607 + 0.E-24*i
> j;
3276.80112729920227590594817065393 + 0.E-25*i
> E;
Elliptic Curve defined by y^2 = x^3 +
(-43201.412594209236689285431925551172 + 0.E-24*i)*x +
(2376116.99598582181541583667180037300 + 0.E-22*i) over Complex
Field
> jInvariant(E);
3276.80112729920227590594817070563 + 0.E-25*i
> w1,w2 := EllipticPeriods(A,100);
> w1;
1.263088700712760693712816573302450091088 + 0.E-38*i
> w2;
0.E-38 - 1.01702927066995984919787906165005620863321*i
> w1/w2;
0.E-38 + 1.2419393788742296224466874060948650840497*i
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