{
  "id": 6271,
  "curve_key": "12685296:0",
  "ainvs": [
    "0",
    "0",
    "0",
    "-264277",
    "0"
  ],
  "rank_lower_bound": 5,
  "torsion": [
    2
  ],
  "family": null,
  "families": [
    "quartic"
  ],
  "naive_height": 49.06786225617147,
  "faltings_height": 2.1572289329891134,
  "conductor": "2234954647328",
  "bad_primes": [
    "2",
    "13",
    "29",
    "701"
  ],
  "discriminant": "1181294218663803712",
  "regulator": "1363.64861943101027371712925258058666874717003837",
  "points": [
    [
      "-468",
      "4602"
    ],
    [
      "-377",
      "6786"
    ],
    [
      "-116",
      "5394"
    ],
    [
      "-1856/9",
      "182584/27"
    ],
    [
      "3393/4",
      "157209/8"
    ]
  ],
  "submitter": "The Wren",
  "history": [],
  "commentary": "Quartic twist y^2 = x^3 + ax (j = 1728) of rank 5. 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 23:32:48",
  "updated_at": "2026-10-31 23:32:48"
}