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Several different models for modular curves are available. The possible
model types are "Atkin", "Canonical", and "Classical", each giving
affine models defined by the modular polynomial databases of the same
names.
For more details on these polynomials see Section Modular Polynomial Databases
on modular polynomials and databases.
Returns a model of the modular curve X0(N), in an affine plane
specified by X. The string t must be one of "Atkin",
"Canonical", or "Classical", with N a level in the corresponding
modular curve database.
Returns an affine model of the modular curve X0(N) of level N from
a database D of modular curves.
Points on modular curves can be created in the same way as points on
curves or schemes in general. In addition, there exist several specific
constructors, defined in terms of the moduli structure, which take a
parameterized elliptic curve as an argument.
Given a modular curve X = X0(N) and an elliptic curve E, with
compatible base rings, returns the sequence of points over the base
field of E, corresponding to E with additional level structure.
Below we give an example of the use of the moduli interpretation of
modular curves in order to construct the corresponding subgroup scheme
structures defined over a finite field.
> FF := FiniteField(NextPrime(10^6));
> A2 := AffineSpace(FF,2);
> X0 := ModularCurve(A2,"Canonical",17);
> E := EllipticCurve([FF|1,23]);
> mp := ModuliPoints(X0,E); mp;
[ (259805, 350390), (380571, 350390) ]
We will see later that it is possible to construct the structure of an
elliptic curve parameterized by the corresponding moduli points.
> P, Q := Explode(mp);
> SubgroupScheme(E,P);
Subgroup of E defined by x^8 + 377217*x^7 + 190510*x^6 + 872850*x^5
+ 816054*x^4 + 457629*x^3 + 64955*x^2 + 361795*x + 460146
> SubgroupScheme(E,Q);
Subgroup of E defined by x^8 + 796070*x^7 + 308587*x^6 + 62023*x^5
+ 976430*x^4 + 380273*x^3 + 200328*x^2 + 892738*x + 536749
In this example we see that the prime 17 splits in the endomorphism
ring of E, so we have exactly two parameterized isogenies defined over
the base field of E.
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