Gonality of the modular curve $X_0(N)$

Filip Najman, Petar Orlić

Published 2022-07-24Version 1

In this paper we determine the $\Q$-gonalities of the modular curves $X_0(N)$ for all $N<150$, except for $N=97,133,135,145$. We also determine the $\mathbb C$-gonality of many of these curves and the $\mathbb Q$-gonalities and $\mathbb C$-gonalities for some larger values of $N$. As a consequence of these results, we determine all the modular curves $X_0(N)$ that are tetragonal over $\mathbb Q$. We find the first known instances of pentagonal curves $X_0(N)$, both over $\mathbb Q$ and over $\mathbb C$. Furthermore, we show that $X_0(109)$ is the only pentagonal curve over $\mathbb Q$, except possibly $X_0(97)$ (whose $\mathbb Q$-gonality is either $5$ or $6$).