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arXiv · 2207.11650

Gonality of the modular curve $X_0(N)$

Filip Najman·Petar Orlić
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Abstract

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 $\mathbb C$.

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Filip Najman, Petar Orlić. 2022-07-24. Gonality of the modular curve $X_0(N)$. https://arxiv.org/abs/2207.11650

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