Invariant theory for the elliptic normal quintic, I. Twists of X(5)

TL;DR

This paper develops invariant theory methods for genus one curves of degree 5, providing algorithms for computing invariants, equations for certain Tate-Shafarevich group elements, and formulas for elliptic curve congruences.

Contribution

It introduces a general approach to invariant theory for elliptic normal quintics, extending previous algorithms and deriving new formulas for 5-congruences and Selmer group elements.

Findings

Algorithm for computing invariants of genus one degree 5 models

Formula for elliptic curves 5-congruent without Weil pairing

Algorithm for doubling elements in the 5-Selmer group

Abstract

A genus one curve of degree 5 is defined by the 4 x 4 Pfaffians of a 5 x 5 alternating matrix of linear forms on P^4. We describe a general method for investigating the invariant theory of such models. We use it to explain how we found our algorithm for computing the invariants [12] and to extend our method in [14] for computing equations for visible elements of order 5 in the Tate-Shafarevich group of an elliptic curve. As a special case of the latter we find a formula for the family of elliptic curves 5-congruent to a given elliptic curve in the case the 5-congruence does not respect the Weil pairing. We also give an algorithm for doubling elements in the 5-Selmer group of an elliptic curve, and make a conjecture about the matrices representing the invariant differential on a genus one normal curve of arbitrary degree.