Gonality and quintic points on intermediate modular curves
For $\{\pm 1\}\subseteq\Delta\subseteq(\mathbb{Z}/N\mathbb{Z})^\times$ let $X_\Delta(N)$ be the intermediate modular curve between $X_1(N)$ and $X_0(N)$. We determine the $\mathbb{Q}$-gonality of every strict intermediate curve $X_\Delta(N)$ with $N\le 50$, and of $X_1(N)$ for $41\le N\le 50$, where it equals the upper bound of Derickx and van Hoeij at every level. We also settle the thirty levels left open by Derickx, Hwang, Jeon and Orli\'c, completing the list of curves $X_0(N)$ with infinitely many points of degree $5$, and we show that exactly $43$ strict intermediate curves have…
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