{
  "id": 6308,
  "curve_key": "-167140113673104:0",
  "ainvs": [
    "0",
    "0",
    "0",
    "3482085701523",
    "0"
  ],
  "rank_lower_bound": 9,
  "torsion": [
    2
  ],
  "family": null,
  "families": [
    "quartic"
  ],
  "naive_height": 98.24956074182619,
  "faltings_height": 6.255703806793672,
  "conductor": "387997466648029537424624928",
  "bad_primes": [
    "2",
    "3",
    "11",
    "13",
    "23",
    "349",
    "877",
    "1153"
  ],
  "discriminant": "-2702080861684501454297700550300066730688",
  "regulator": "767764487.82495193572714733898756450022317505887401680413",
  "points": [
    [
      "9238801/9",
      "58261273642/27"
    ],
    [
      "13937664/25",
      "181769351712/125"
    ],
    [
      "28420297/16",
      "219749175803/64"
    ],
    [
      "62773932/49",
      "878740887102/343"
    ],
    [
      "170404608/361",
      "9070605332976/6859"
    ],
    [
      "382448209/2025",
      "74275253211302/91125"
    ],
    [
      "931855753/1089",
      "68244121266274/35937"
    ],
    [
      "1006674097/144",
      "33058170671383/1728"
    ],
    [
      "6907377411/841",
      "588706566902982/24389"
    ]
  ],
  "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 23:58:03",
  "updated_at": "2026-10-31 23:58:03"
}