Twists of non-hyperelliptic curves of genus 3

TL;DR

This paper explicitly computes equations for all twists of smooth plane quartic curves of genus 3 over a number field, leveraging a classification of their automorphism groups to facilitate the process.

Contribution

It provides a comprehensive method to determine twists of non-hyperelliptic genus 3 curves, extending previous work with explicit equations.

Findings

Explicit equations for all twists of plane quartic curves over number fields

Application of automorphism group classification to compute twists

Extension of methods to non-hyperelliptic genus 3 curves

Abstract

In this paper we explicitly compute equations for the twists of all the smooth plane quartic curves defined over a number field k. Since the plane quartic curves are non-hyperelliptic curves of genus 3 we can apply the method developed by the author in a previous article. The starting point is a classification due to Henn of the plane quartic curves with non-trivial automorphism group up to C-isomorphism.