Plane quartic twists of X(5,3)

TL;DR

This paper develops a method to explicitly construct plane quartic models for certain genus-three modular curves associated with quadratic Q-curves of degree 5, linked to specific Galois representations.

Contribution

It introduces a new explicit construction technique for plane quartic models of twisted modular curves related to PGL(2,3) representations.

Findings

Explicit plane quartic models for the genus-three curves are obtained.

The method applies to quadratic Q-curves of degree 5.

Provides a practical approach for studying rational points on these curves.

Abstract

Given an odd representation of the absolute Galois group of Q onto PGL(2,3) and a positive integer N, there exists a twisted modular curve defined over Q whose rational points classify the quadratic Q-curves of degree N realizing the representation. The paper gives a method to provide an explicit plane quartic model for this curve in the genus-three case N=5.