{
  "id": "2311.07740",
  "version": "v1",
  "published": "2023-11-13T20:44:09.000Z",
  "updated": "2023-11-13T20:44:09.000Z",
  "title": "Towards a Classification of Isolated $j$-invariants",
  "authors": [
    "Abbey Bourdon",
    "Sachi Hashimoto",
    "Timo Keller",
    "Zev Klagsbrun",
    "David Lowry-Duda",
    "Travis Morrison",
    "Filip Najman",
    "Himanshu Shukla"
  ],
  "comment": "With an appendix by Maarten Derickx and Mark van Hoeij",
  "categories": [
    "math.NT"
  ],
  "abstract": "We develop an algorithm to test whether a non-CM elliptic curve $E/\\mathbb{Q}$ gives rise to an isolated point of any degree on any modular curve of the form $X_1(N)$. This builds on prior work of Zywina which gives a method for computing the image of the adelic Galois representation associated to $E$. Running this algorithm on all elliptic curves presently in the $L$-functions and Modular Forms Database and the Stein-Watkins Database gives strong evidence for the conjecture that $E$ gives rise to an isolated point on $X_1(N)$ if and only if $j(E)=-140625/8, -9317,$ $351/4$, or $-162677523113838677$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-13T20:44:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "classification",
      "invariants",
      "non-cm elliptic curve",
      "isolated point",
      "adelic galois representation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}