{
  "id": 6288,
  "curve_key": "15857183328:0",
  "ainvs": [
    "0",
    "0",
    "0",
    "-330357986",
    "0"
  ],
  "rank_lower_bound": 7,
  "torsion": [
    2
  ],
  "family": null,
  "families": [
    "quartic"
  ],
  "naive_height": 70.46066532424918,
  "faltings_height": 3.939962521995588,
  "conductor": "6984729530494476544",
  "bad_primes": [
    "2",
    "7",
    "113",
    "367",
    "569"
  ],
  "discriminant": "2307461180448880855200080384",
  "regulator": "196415.8409043041072332830452136312195951911880878322",
  "points": [
    [
      "-6606",
      "1376250"
    ],
    [
      "-2569",
      "911995"
    ],
    [
      "18306",
      "294930"
    ],
    [
      "19775",
      "1095535"
    ],
    [
      "56952",
      "12880644"
    ],
    [
      "-373239/256",
      "2833506075/4096"
    ],
    [
      "-82208/9",
      "40549096/27"
    ]
  ],
  "submitter": "The Wren",
  "history": [],
  "commentary": "Quartic twist y^2 = x^3 + ax (j = 1728) of rank 7. 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 23:45:19",
  "updated_at": "2026-10-31 23:45:19"
}