{
  "id": "2207.11650",
  "version": "v1",
  "published": "2022-07-24T03:46:09.000Z",
  "updated": "2022-07-24T03:46:09.000Z",
  "title": "Gonality of the modular curve $X_0(N)$",
  "authors": [
    "Filip Najman",
    "Petar Orlić"
  ],
  "comment": "17 pages",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "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 $6$).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-24T03:46:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "modular curve",
      "pentagonal curve",
      "larger values",
      "consequence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}