Computing minimal Weierstrass equations

TL;DR

This paper presents algorithms for computing minimal Weierstrass equations of hyperelliptic curves over principal ideal domains, including special cases with rational Weierstrass points, improving the process of simplifying such equations.

Contribution

The paper introduces new algorithms for finding minimal Weierstrass equations of hyperelliptic curves, extending to cases with rational Weierstrass points.

Findings

Algorithms successfully compute minimal Weierstrass equations

Applicable to hyperelliptic curves over principal ideal domains

Includes methods for curves with rational Weierstrass points

Abstract

We describe an algorithm for determining a minimal Weierstrass equation for hyperelliptic curves over principal ideal domains. When the curve has a rational Weierstrass point $w_0$, we also give a similar algorithm for determining the minimal Weierstrass equation with respect to $w_0$.