A formula for the Jacobian of a genus one curve of arbitrary degree

TL;DR

This paper generalizes classical invariant theory formulas for the Jacobian of genus one curves to arbitrary degrees by associating a quadratic form matrix and identifying key invariants.

Contribution

It introduces a new method to compute the Jacobian of genus one curves of any degree using invariant polynomials and quadratic form matrices.

Findings

Extended classical formulas to arbitrary degree

Constructed an $n imes n$ alternating matrix for each curve

Identified invariants as homogeneous polynomials of degrees 4 and 6

Abstract

We extend the formulae of classical invariant theory for the Jacobian of a genus one curve of degree $n \le 4$ to curves of arbitrary degree. To do this, we associate to each genus one normal curve of degree $n$, an $n \times n$ alternating matrix of quadratic forms in $n$ variables, that represents the invariant differential. We then exhibit the invariants we need as homogeneous polynomials of degrees $4$ and $6$ in the coefficients of the entries of this matrix.