{
  "id": 6309,
  "curve_key": "169684324811616:0",
  "ainvs": [
    "0",
    "0",
    "0",
    "-3535090100242",
    "0"
  ],
  "rank_lower_bound": 9,
  "torsion": [
    2
  ],
  "family": null,
  "families": [
    "quartic"
  ],
  "naive_height": 98.29488274171797,
  "faltings_height": 6.259480640117987,
  "conductor": "799799169077055590941348096",
  "bad_primes": [
    "2",
    "41",
    "881",
    "4889",
    "10009"
  ],
  "discriminant": "2827362124786076755603056787831255839232",
  "regulator": "2988567991.2726184555393340447102280804068892989213821980",
  "points": [
    [
      "-696200",
      "1457287020"
    ],
    [
      "-570888",
      "1353543732"
    ],
    [
      "-39147200800/177241",
      "65478291176733960/74618461"
    ],
    [
      "-1081065409/625",
      "15145745126889/15625"
    ],
    [
      "-551322752/361",
      "9295921836816/6859"
    ],
    [
      "-361535089/1225",
      "43250804232159/42875"
    ],
    [
      "-47161472/361",
      "4649970671184/6859"
    ],
    [
      "-156800/9",
      "6700344560/27"
    ],
    [
      "1953766809/484",
      "76419776544531/10648"
    ]
  ],
  "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": "2026-10-31 23:58:38",
  "updated_at": "2026-10-31 23:58:38"
}