On hyperelliptic curves of genus 3

TL;DR

This paper explores the moduli space of genus 3 hyperelliptic curves using weighted projective space, resulting in a comprehensive database of such curves over rationals with minimal height.

Contribution

It introduces a novel approach to classify genus 3 hyperelliptic curves via weighted projective space and constructs a database of these curves over ields with height =1.

Findings

Created a database of all genus 3 hyperelliptic curves over ields with height =1

Utilized weighted projective space of binary octavics for classification

Enhanced understanding of the moduli space structure for genus 3 hyperelliptic curves

Abstract

We study the moduli space of genus 3 hyperelliptic curves via the weighted projective space of binary octavics. This enables us to create a database of all genus 3 hyperelliptic curves defined over $\mathbb Q$, of weighted moduli height $\mathcal h =1$.