#### Vol. 12, No. 9, 2018

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

### Tom Fisher

Vol. 12 (2018), No. 9, 2123–2150
##### 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 $n×n$ 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.

##### Keywords
elliptic curves, invariant theory, higher secant varieties
##### Mathematical Subject Classification 2010
Primary: 11G05
Secondary: 13D02, 14H52