Associated Structures

FunctionField(X): CrvMod -> FldFun

Returns the function field of the modular curve \(X\).

jFunction(X): CrvMod -> FldFunElt

Given a modular curve \(X\) over a field, returns the \(j\)-invariant as a function on the curve.

BaseCurve(X): CrvMod -> CrvMod, MapSch

Given one of the standard models \(X\) for the modular curve \(X_0(N)\), returns the base model curve \(X(1)\) and the morphism \(\pi: X_0(N) \rightarrow X(1)\).

Example: Modular base curve (ex-d78d4a)

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);

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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 \(X_0(17)\) 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) ]

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We note that \(X(1)\) is defined to be the classical modular curve, defined by the diagonal image of the \(j\)-line in \({\mathbb{P}}^2\).