{
  "id": 477,
  "curve_key": "148614550896:-16988577803075160",
  "ainvs": [
    "0",
    "0",
    "1",
    "-3096136477",
    "19662705790596"
  ],
  "rank_lower_bound": 13,
  "torsion": [],
  "naive_height": 77.17386565298568,
  "faltings_height": 4.4938248892284705,
  "conductor": "1732483513617389907070245271237",
  "bad_primes": [
    "1732483513617389907070245271237"
  ],
  "discriminant": "1732483513617389907070245271237",
  "regulator": "13064571333.16078945405592950290560539378515472496571595340731174",
  "points": [
    [
      "6431",
      "131999"
    ],
    [
      "6420",
      "223872"
    ],
    [
      "6327",
      "571599"
    ],
    [
      "6297",
      "644999"
    ],
    [
      "6261",
      "723344"
    ],
    [
      "5998",
      "1143618"
    ],
    [
      "5933",
      "1225631"
    ],
    [
      "5401",
      "1760120"
    ],
    [
      "4856",
      "2177699"
    ],
    [
      "4792",
      "2221724"
    ],
    [
      "4222",
      "2581874"
    ],
    [
      "3986",
      "2717505"
    ],
    [
      "3859",
      "2787863"
    ]
  ],
  "submitter": "NDElkies",
  "history": [],
  "commentary": "prime conductor, the smallest among the rank-13 curves found in the searches earlier this year that found Curves 157, 158, 244  of ranks 12, 13, 14 and smallest conductor known.  The 13 points listed are the 13 integral points of minimal height (with y>0); they are independent but not saturated at 2: replacing the 11th point [4222, 2581874] by  [1269, 3966831] makes the regulator 4 times smaller.  Evidently the code that certifies independence can recognize independent points even when they are dependent mod 2.",
  "created_at": "2026-09-01 04:50:30",
  "updated_at": "2026-09-01 04:50:30"
}