Minimal Weierstrass equations for genus 2 curves

Abstract

We study the minimal Weierstrass equations for genus 2 curves defined over a ring of integers \(\mathcal O_{\mathbb F}\). This is done via reduction theory and Julia invariant of binary sextics. We show that when the binary sextics has extra automorphisms this is usually easier to compute. Moreover, we show that when the standard form \(y^2=f(x^2)\) such that \(f(0)=1\) is defined over \(\mathcal O_{\mathbb F}\) and the discriminant of \(f(x)\) is minimal then this form has minimal height.