Non-hyperelliptic modular curves of genus 3

Abstract

A curve $C$ defined over $\mathbb Q$ is modular of level $N$ if there exists a non-constant morphism from $X_1(N)$ onto $C$ defined over $\mathbb Q$ for some positive integer $N$. We provide a sufficient and necessary condition for the existence of a modular non-hyperelliptic curve $C$ of genus $3$ and level $N$ such that $\mathrm{Jac}{(C)}$ is $\mathbb Q$-isogenous to a given three dimensional $\mathbb Q$-quotient of $J_1 (N)$. Using this criterion, we present an algorithm to compute explicitly equations for modular non-hyperelliptic curves of genus $3$. Let $C$ be a modular curve of level $N$, we say that $C$ is new if the corresponding morphism between $J_1(N)$ and $\mathrm{Jac}{(C)}$ factors through the new part of $J_1(N)$.