{
  "id": "1405.6385",
  "version": "v2",
  "published": "2014-05-25T12:43:30.000Z",
  "updated": "2014-12-08T14:56:22.000Z",
  "title": "Families Of Elliptic Curves With The Same Mod 8 Representations",
  "authors": [
    "Zexiang Chen"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $E:y^2=x^3+ax+b$ be an elliptic curve defined over $\\mathbb{Q}$. We compute certain twists of the classical modular curves $X(8)$. Searching for rational points on these twists enables us to find non-trivial pairs of $8$-congruent elliptic curves over $\\mathbb{Q}$, i.e. pairs of non-isogenous elliptic curves over $\\mathbb{Q}$ whose $8$-torsion subgroups are isomorphic as Galois modules. We also show that there are infinitely many examples over $\\mathbb{Q}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-25T12:43:30.000Z",
      "abstract": "Let $E:y^2=x^3+ax+b$ be an elliptic curve defined over $\\mathbb{Q}$. We compute certain twists of the classical modular curves $X(8)$ which parameterize families of elliptic curves which have the same mod 8 representation as $E$ together with some specified conditions on the Weil pairing.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-12-08T14:56:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F33",
      "11F80"
    ],
    "keywords": [
      "representation",
      "classical modular curves",
      "parameterize families",
      "elliptic curve"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.6385C"
    }
  }
}