Returns the function field of the modular curve X.
Given a modular curve X over a field, returns the j-invariant as a function on the curve.
Given one of the standard models X for the modular curve X0(N), returns the base model curve X(1) and the morphism π: X0(N) -> 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.