# Some minimisation algorithms in arithmetic invariant theory

## TL;DR

This paper develops algorithms for minimizing various representations of genus one curves, extending previous work and establishing a link between minimal discriminants and Jacobian elliptic curves.

## Contribution

It introduces new minimization algorithms for bidegree (2,2)-forms, 3x3x3 cubes, and 2x2x2x2 hypercubes, expanding the scope of prior methods.

## Key findings

- Algorithms for minimizing specific genus one curve representations.
- A theorem relating minimal discriminant to Jacobian elliptic curves.
- Extension of previous minimization techniques to new algebraic forms.

## Abstract

We extend the work of Cremona, Fisher and Stoll on minimising genus one curves of degrees 2,3,4,5, to some of the other representations associated to genus one curves, as studied by Bhargava and Ho. Specifically we describe algorithms for minimising bidegree (2,2)-forms, 3 x 3 x 3 cubes and 2 x 2 x 2 x 2 hypercubes. We also prove a theorem relating the minimal discriminant to that of the Jacobian elliptic curve.