On genus one curves of degree 5 with square-free discriminant

TL;DR

This paper investigates genus one degree 5 curves defined by Pfaffians, providing new invariant formulas, establishing minimality criteria, and demonstrating integrality of transformations for models with square-free discriminant, impacting elliptic curve rank studies.

Contribution

It introduces new formulas for invariants, proves equivalence of minimality definitions, and shows transformations are integral for models with square-free discriminant.

Findings

New invariant formulas for genus one degree 5 curves

Equivalence of two minimality definitions

Transformations are integral for square-free discriminant models

Abstract

We study genus one curves of degree 5 defined by Pfaffians. We give new formulae for the invariants, and prove the equivalence of two different definitions of minimality. As an application we show that transformations between models with square-free discriminant are necessarily integral. This result is used by Bhargava and Shankar in their work on the average ranks of elliptic curves.