Gonality of the modular curve $X_0(N)$

In this paper we determine the $\mathbb{Q}$-gonalities of the modular curves $X_0(N)$ for all $N<145$. We determine the $\mathbb{C}$-gonality of many of these curves and the $\mathbb{Q}$-gonalities and $\mathbb{C}$-gonalities for many larger values of $N$. Using these results and some further work, we determine all the modular curves $X_0(N)$ of gonality $4$, $5$ and $6$ over $\mathbb{Q}$. We also find the first known instances of pentagonal curves $X_0(N)$ over $\mathbb{C}$.