{
  "id": "1702.07407",
  "version": "v1",
  "published": "2017-02-23T22:07:27.000Z",
  "updated": "2017-02-23T22:07:27.000Z",
  "title": "Binary quartic forms with bounded invariants and small Galois groups",
  "authors": [
    "Cindy",
    "Tsang",
    "Stanley Yao Xiao"
  ],
  "comment": "45 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper, we enumerate the $\\operatorname{GL}_2(\\mathbb{Z})$-equivalence classes of integral binary quartic forms which are fixed under substitution by a particular matrix in $\\operatorname{GL}_2(\\mathbb{R})$ which is proportional over $\\mathbb{R}$ to an integer matrix. In particular, whenever such a form $F$ is irreducible, the Galois group of the splitting field of $F$ is isomorphic to a subgroup of the dihedral group $\\mathcal{D}_4$ of order eight. We also give a new criterion for when the negative Pell's equation $x^2 - Dy^2 = -1$ is soluble in integers $x$ and $y$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-02-23T22:07:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E20",
      "14G25"
    ],
    "keywords": [
      "small galois groups",
      "bounded invariants",
      "integral binary quartic forms",
      "integer matrix",
      "equivalence classes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 45,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}