{
  "id": 6269,
  "curve_key": "8946528:0",
  "ainvs": [
    "0",
    "0",
    "0",
    "-186386",
    "0"
  ],
  "rank_lower_bound": 5,
  "torsion": [
    2
  ],
  "family": null,
  "families": [
    "quartic"
  ],
  "naive_height": 48.020328246074065,
  "faltings_height": 2.069934432147662,
  "conductor": "2223343423744",
  "bad_primes": [
    "2",
    "41",
    "2273"
  ],
  "discriminant": "414400087377949184",
  "regulator": "2966.18685790221646762225682133333278864308078959",
  "points": [
    [
      "-369",
      "4305"
    ],
    [
      "-328",
      "5084"
    ],
    [
      "-25",
      "2155"
    ],
    [
      "-20992/49",
      "379168/343"
    ],
    [
      "2601/4",
      "99195/8"
    ]
  ],
  "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 23:30:41",
  "updated_at": "2026-10-31 23:30:41"
}