{
  "id": 6317,
  "curve_key": "850548883077696:0",
  "ainvs": [
    "0",
    "0",
    "0",
    "-17719768397452",
    "0"
  ],
  "rank_lower_bound": 9,
  "torsion": [
    2
  ],
  "family": null,
  "families": [
    "quartic"
  ],
  "naive_height": 103.13070900542021,
  "faltings_height": 6.6624661620931755,
  "conductor": "627980384118677240480184608",
  "bad_primes": [
    "2",
    "11",
    "97",
    "881",
    "1697",
    "2777"
  ],
  "discriminant": "356085342871228953501618765896360858202112",
  "regulator": "2176803427.1504709135016737526704590941302220708360576306",
  "points": [
    [
      "-2357003",
      "5354559177"
    ],
    [
      "-37477115563058/13271449",
      "253630529263292627052/48347888707"
    ],
    [
      "-321811911872/2954961",
      "7054015978840934224/5079577959"
    ],
    [
      "-41767279664/403225",
      "346787186470924584/256047875"
    ],
    [
      "-10764762800/3249",
      "875286092696840/185193"
    ],
    [
      "-6889448192/1681",
      "134026463946144/68921"
    ],
    [
      "-414551907/169",
      "11771161918131/2197"
    ],
    [
      "11628390361/36",
      "1253841157866563/216"
    ],
    [
      "10068050225825/872356",
      "29745391914495103215/814780504"
    ]
  ],
  "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 00:03:53",
  "updated_at": "2026-11-01 00:03:53"
}