{
  "id": 6307,
  "curve_key": "130327764063216:0",
  "ainvs": [
    "0",
    "0",
    "0",
    "-2715161751317",
    "0"
  ],
  "rank_lower_bound": 9,
  "torsion": [
    2
  ],
  "family": null,
  "families": [
    "quartic"
  ],
  "naive_height": 97.50322096608014,
  "faltings_height": 6.193508825481501,
  "conductor": "235907306746073553639503648",
  "bad_primes": [
    "2",
    "97",
    "101",
    "197",
    "829",
    "1697"
  ],
  "discriminant": "1281052992266291597026264928356580608832",
  "regulator": "56176585.564549740277813725875828896866265817630235340333",
  "points": [
    [
      "-1481481",
      "878024406"
    ],
    [
      "-321652",
      "916547374"
    ],
    [
      "2312189",
      "2466475066"
    ],
    [
      "-277142161/169",
      "452850291074/2197"
    ],
    [
      "-202581072/361",
      "7960406342484/6859"
    ],
    [
      "-8121713/9",
      "35361938402/27"
    ],
    [
      "44292641/16",
      "236871266767/64"
    ],
    [
      "1599977461/729",
      "42274604474542/19683"
    ],
    [
      "27991358261/1369",
      "4667894886320882/50653"
    ]
  ],
  "submitter": "The Wren",
  "history": [],
  "commentary": "Quartic twist y^2 = x^3 + ax (j = 1728) of rank 9. Rank proven (independent generators plus a matching 2-descent upper bound) in M. Rubinstein's elliptic-curves-by-rank collection, file families/family_Z2.add_2026-10-24.part1.tsv.gz:\r\nhttps://github.com/mrubinst/elliptic-curves-by-rank",
  "created_at": "2026-10-31 23:57:25",
  "updated_at": "2026-10-31 23:57:25"
}