Standard Functions and Forms

This section contains intrinsics that return the \(j\)-invariant as a rational function and normalised Eisenstein forms as meromorphic \(k\)-differentials on the small database models of \(X_0(N)\). Standard variants of these can be obtained by pulling back by Atkin-Lehner involutions or pulling back the corresponding objects from lower level curves by the projection or \(r\)-projection maps.

The database actually only contains precomputed expressions for \(E_2^{(N)}\) (see below), \(E_4\) and \(E_6\) for prime levels \(N\) and reconstructs everything else from these objects using projection maps.

jInvariant(CN, N): Crv, RngIntElt -> FldFunRatMElt

jFunction(CN, N): Crv, RngIntElt -> FldFunFracSchElt

The curve \(CN\) should be a base change of the small modular database curve of level \(N\) to a field of characteristic 0. These intrinsics return the \(j\)-invariant \(j(z)\) as a rational function on \(CN\). The second returns it as an element of the function field of \(CN\). The first returns it as an element in the field of fractions of the coordinate ring of the ambient of \(CN\), which is sometimes more convenient to use.

jInvariant(p, N): Pt, RngIntElt -> RngElt

jInvariant(p, N): PlcCrvElt, RngIntElt -> RngElt

The point or place \(p\) is a non-cuspidal point or a non-cuspidal place on a base change \(CN\) of the small modular database curve of level \(N\) to a field of characteristic 0. Returns the value of \(j(z)\) at \(p\), the value lying in \(L\) if \(p\) is a point in \(CN(L)\) or in the residue class field of \(p\) if \(p\) is a place of \(CN\). If \(p\) is a point, it should be non-singular. However, if the \(j\) function is defined at \(p\), the intrinsic will still return a value.

jNInvariant(p, N): Pt, RngIntElt -> RngElt

jNInvariant(p, N): PlcCrvElt, RngIntElt -> RngElt

Exactly as jInvariant above, except that the intrinsic gives the value of rational function \(j(Nz)\) at the point or place. This is equivalent to computing the \(j\)-invariant value on the image of \(p\) under the Fricke involution \(w_N\).

E2NForm(CN, N): Crv, RngIntElt -> DiffCrvElt

The curve \(CN\) should be a base change of the small modular database curve of level \(N\) to a field of characteristic 0. \(E_2^{(N)}(z)=NE_2(Nz)-E_2(z)\) is a weight 2 integral form for \(\Gamma_0(N)\) where \(E_2(z)=1-24e^{2\pi iz}+\ldots\) is the normalised weight 2 Eisenstein series. \(E_2^{(N)}(z)\) corresponds to a meromorphic differential (defined over Q) on \(X_0(N)\). The intrinsic returns \(E_2^{(N)}(z)\) as a meromorphic differential in the function field of \(CN\).

E4Form(CN, N): Crv, RngIntElt -> FldFunFracSchElt, DiffCrvElt

The curve \(CN\) should be a base change of the small modular database curve of level \(N\) to a field of characteristic 0. Returns a rational function \(f\) and a differential form \(\omega\) in the function field of \(CN\) such that the Eisenstein series \(E_4(z)=1+240e^{2\pi iz}+\ldots\) as a meromorphic 2-differential on \(CN\) is given by \(f\omega^2\).

E6Form(CN, N): Crv, RngIntElt -> FldFunFracSchElt, DiffCrvElt

The curve \(CN\) should be a base change of the small modular database curve of level \(N\) to a field of characteristic 0. Returns a rational function \(f\) and a differential form \(\omega\) in the function field of \(CN\) such that the Eisenstein series \(E_6(z)=1-504e^{2\pi iz}+\ldots\) as a meromorphic 3-differential on \(CN\) is given by \(f\omega^3\). Note that the same differential \(\omega\) is returned by the E4Form intrinsic.