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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.
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) : 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) : 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.
The curve 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 - 24e2π iz + ... 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.
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 ω in the function field of CN such that the Eisenstein
series E4(z)=1 + 240e2π iz + ... as a meromorphic 2-differential on CN is given by fω2.
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 ω in the function field of CN such that the Eisenstein
series E6(z)=1 - 504e2π iz + ... as a meromorphic 3-differential on CN is given by fω3.
Note that the same differential ω is returned by the E4Form intrinsic.
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