{
  "id": 6316,
  "curve_key": "668560454692416:0",
  "ainvs": [
    "0",
    "0",
    "0",
    "-13928342806092",
    "0"
  ],
  "rank_lower_bound": 9,
  "torsion": [
    2
  ],
  "family": null,
  "families": [
    "quartic"
  ],
  "naive_height": 102.40844382518586,
  "faltings_height": 6.602277397073645,
  "conductor": "387997466648029537424624928",
  "bad_primes": [
    "2",
    "3",
    "11",
    "13",
    "23",
    "349",
    "877",
    "1153"
  ],
  "discriminant": "172933175147808093075052835219204270764032",
  "regulator": "15355289.756499038714542946779751290004463501177480336083",
  "points": [
    [
      "-2525952",
      "4366423776"
    ],
    [
      "-1420848",
      "4113591768"
    ],
    [
      "-752466",
      "3170891724"
    ],
    [
      "-583418",
      "2815575268"
    ],
    [
      "6067086",
      "11782281012"
    ],
    [
      "20035392",
      "88110645168"
    ],
    [
      "45824832",
      "309176322768"
    ],
    [
      "183031654/49",
      "103595916164/343"
    ],
    [
      "211035497062/27889",
      "84335271548898812/4657463"
    ]
  ],
  "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 00:03:19",
  "updated_at": "2026-11-01 00:03:19"
}