The functions in this section implement the invariant theory developed in [Fis].
We write Xn for the (affine) space of genus one models of degree n. The module of covariants Xn to Xn is a free module of rank 2 over the ring of invariants. The generators are the identity map and a second covariant which we term the Hessian. (In the cases where n=2 or 3 this is the determinant of a matrix of second partial derivatives.) This function evaluates the Hessian of the given genus one model.
The covariants that define the covering map from the given genus one model to its Jacobian (this is the same as the defining equations of the nCovering).
We write Xn for the (affine) space of genus one models of degree n, and Xn * for its dual. The module of contravariants Xn to Xn * is a free module of rank 2 over the ring of invariants. This function evaluates the generators P and Q at the given genus one model.
Evaluates a pair of covariants, which depend on an integer r, at the genus one model of degree prime to r. The pencil spanned by these genus one models is a family of genus one curves invariant under the same representation of the Heisenberg group. (In other words, the universal family above a twist of X(n).)If r ≡ 1 mod (n) then the covariants evaluated are the identity map and the Hessian. If r ≡ - 1 mod (n) then the covariants evaluated are the contravariants. If n=5 there are two further possibilities. We identify X5= ^2 V tensor W where V and W are 5-dimensional vector spaces. Then the covariants evaluated for r ≡ 2, 3 mod (5) take values in ^2 W tensor V * and ^2 W * tensor V respectively.
Variables: [ RngMPolElt ] Default:
The Hesse polynomials D(x, y), c4(x, y), c6(x, y). These polynomials give the invariants for the pencil of genus one models computed by HesseCovariants. The RubinSilverbergPolynomials are closely related to these formulae in the case r ≡ 1 mod (n).