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# Plane Quartic Twists of $X(5,3)$

Published:2007-06-01
Printed: Jun 2007
• Julio Fernández
• Josep González
• Joan-C. Lario
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## Abstract

Given an odd surjective Galois representation $\varrho\from \G_\Q\to\PGL_2(\F_3)$ and a positive integer~$N$, there exists a twisted modular curve $X(N,3)_\varrho$ defined over $\Q$ whose rational points classify the quadratic $\Q$-curves of degree $N$ realizing~$\varrho$. This paper gives a method to provide an explicit plane quartic model for this curve in the genus-three case $N=5$.
 MSC Classifications: 11F03 - Modular and automorphic functions 11F80 - Galois representations 14G05 - Rational points