Example CrvMod_Modular base curve (H137E4)

In this example we demonstrate the relation of a modular curve with its base curve X(1).

> D := ModularCurveDatabase("Atkin");
> X0 := ModularCurve(D,17);
> X1, pi := BaseCurve(X0);

The discriminants -4, -8, -16, -19, -43, and -67 are the class number 1 discriminants in which 17 is a split prime. We use this to construct the corresponding moduli points on the curve X₀(17) and map these back down to the curve X(1). Refer to Section Class Polynomials for a description of the function HilbertClassPolynomial.

> discs := [ -4, -8, -16, -19, -43, -67 ];
> jinvs := [ Roots(HilbertClassPolynomial(D))[1][1] : D in discs ];
> jinvs;
[ 1728, 8000, 287496, -884736, -884736000, -147197952000 ]
> pnts := &cat[ ModuliPoints(X0, EllipticCurveFromjInvariant(j))
> : j in jinvs ];
> pnts;
[ (-2, 1728), (-3, 8000), (-4, 287496), (-1, -884736), (2, -884736000),
(7, -147197952000) ]
> [ pi(P) : P in pnts ];
[ (1728, 1728), (8000, 8000), (287496, 287496), (-884736, -884736),
(-884736000, -884736000), (-147197952000, -147197952000) ]

We note that X(1) is defined to be the classical modular curve, defined by the diagonal image of the j-line in P².