Example ModFrm_Relations (H144E20)

We compute an equation that defines the canonical embedding of X₀(34).

> S := CuspidalSubspace(ModularForms(Gamma0(34)));
> Relations(S, 4, 20);
[
    a^3*c - a^2*b^2 - 3*a^2*c^2 + 2*a*b^3 + 3*a*b^2*c - 3*a*b*c^2 + 
        4*a*c^3 - b^4 + 4*b^3*c - 6*b^2*c^2 + 4*b*c^3 - 2*c^4
]
[
    (0  0  1 -1  0 -3  2  3 -3  4 -1  4 -6  4 -2)
]
> // a, b, and c correspond to the cusp forms S.1, S.2 and S.3:
> S.1;
q - 2*q^4 - 2*q^5 + 4*q^7 + O(q^8)
> S.2;
q^2 - q^4 + O(q^8)
> S.3;
q^3 - 2*q^4 - q^5 + q^6 + 4*q^7 + O(q^8)

Next we compute the canonical embedding of X₀(75).

> S := CuspidalSubspace(ModularForms(Gamma0(75)));
> R := Relations(S, 2, 20); R;
[
    a*c - b^2 - d^2 - 4*e^2,
    a*d - b*c + b*e + d*e - 3*e^2,
    a*e - b*d - c*e
]
> // NOTE: It is much faster to compute in the power 
> // series ring than the ring of modular forms!
> a, b, c, d, e := Explode([PowerSeries(f,20) : f in Basis(S)]);
> a*c - b^2 - d^2 - 4*e^2;
O(q^21)

The connection between the above computations and models for modular curves is discussed in Steven Galbraith's Oxford Ph.D. thesis.

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