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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 X0(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 E2(N) (see below), E4 and E6 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
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
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 wN.
E2NForm(CN,N) : Crv, RngIntElt -> DiffCrvElt
CN should be a base change of the small modular database curve of level N to a field of characteristic 0. E2(N)(z)=NE2(Nz) - E2(z) is a weight 2 integral form for Γ0(N) where E2(z)=1 - 24e2πiz + ... is the normalised weight 2 Eisenstein series. E2(N)(z) corresponds to a meromorphic differential (defined over Q) on X0(N). The intrinsic returns E2(N)(z) as a meromorphic differential in the function field of CN.
E4Form(CN,N) : Crv, RngIntElt -> FldFunFracSchElt, DiffCrvElt
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 ωin the function field of CN such that the Eisenstein series E4(z)=1 + 240e2πiz + ... as a meromorphic 2-differential on CN is given by fω2.
E6Form(CN,N) : Crv, RngIntElt -> FldFunFracSchElt, DiffCrvElt
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 ωin the function field of CN such that the Eisenstein series E6(z)=1 - 504e2πiz + ... as a meromorphic 3-differential on CN is given by fω3. Note that the same differential ωis returned by the E4Form intrinsic.
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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013