Minimisation and reduction of 5-coverings of elliptic curves

TL;DR

This paper develops methods for minimizing and reducing genus one models of degree 5, aiding explicit 5-descent on elliptic curves by providing algorithms for minimal models with preserved invariants.

Contribution

It introduces a theorem on the existence of minimal models for degree 5 genus one curves and provides algorithms for their computation and reduction.

Findings

Established a theorem on minimal models with invariant preservation

Developed algorithms for computing minimal models

Outlined procedures for reducing models over Q

Abstract

We consider models for genus one curves of degree 5, which arise in explicit 5-descent on elliptic curves. We prove a theorem on the existence of minimal models with the same invariants as the minimal model of the Jacobian elliptic curve and give an algorithm for computing such models. Finally we describe how to reduce genus one models of degree 5 defined over Q.