{
  "id": 6312,
  "curve_key": "-212637220769424:0",
  "ainvs": [
    "0",
    "0",
    "0",
    "4429942099363",
    "0"
  ],
  "rank_lower_bound": 9,
  "torsion": [
    2
  ],
  "family": null,
  "families": [
    "quartic"
  ],
  "naive_height": 98.97182592206055,
  "faltings_height": 6.315892571813203,
  "conductor": "627980384118677240480184608",
  "bad_primes": [
    "2",
    "11",
    "97",
    "881",
    "1697",
    "2777"
  ],
  "discriminant": "-5563833482362952398462793217130638409408",
  "regulator": "799642075.27976482536796178669527068763967341377651096635",
  "points": [
    [
      "709676",
      "1871161358"
    ],
    [
      "7064739",
      "19593399474"
    ],
    [
      "9346529",
      "29289844054"
    ],
    [
      "45523361",
      "307478787382"
    ],
    [
      "67222067",
      "551417527078"
    ],
    [
      "237314089/9",
      "3667451931406/27"
    ],
    [
      "2135826801/10609",
      "1036654863325098/1092727"
    ],
    [
      "2847499699/3249",
      "395277599668094/185193"
    ],
    [
      "17552699278796/2550409",
      "76900704697976648918/4073003173"
    ]
  ],
  "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 00:00:26",
  "updated_at": "2026-11-01 00:00:26"
}