{
  "id": 6315,
  "curve_key": "647897065979760:0",
  "ainvs": [
    "0",
    "0",
    "0",
    "-13497855541245",
    "0"
  ],
  "rank_lower_bound": 9,
  "torsion": [
    2
  ],
  "family": null,
  "families": [
    "quartic"
  ],
  "naive_height": 102.31425885259905,
  "faltings_height": 6.594428649358078,
  "conductor": "647794148310465251186311200",
  "bad_primes": [
    "2",
    "3",
    "5",
    "149",
    "601",
    "1201",
    "2789"
  ],
  "discriminant": "157388973018452977416285085934598097992000",
  "regulator": "84500172.880118561441148939201196819114347296372680661562",
  "points": [
    [
      "-3609005",
      "1306459810"
    ],
    [
      "-2077805",
      "4367546110"
    ],
    [
      "-1624500",
      "4200025050"
    ],
    [
      "-1610541",
      "4190627682"
    ],
    [
      "-1620",
      "147873330"
    ],
    [
      "6443505",
      "13436994330"
    ],
    [
      "-5269869101/1521",
      "134934245492662/59319"
    ],
    [
      "-6704756/9",
      "83839621402/27"
    ],
    [
      "537741745/144",
      "2233241466985/1728"
    ]
  ],
  "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": "2026-11-01 00:02:32",
  "updated_at": "2026-11-01 00:02:32"
}