{
  "id": 6314,
  "curve_key": "-521311056252864:0",
  "ainvs": [
    "0",
    "0",
    "0",
    "10860647005268",
    "0"
  ],
  "rank_lower_bound": 9,
  "torsion": [
    2
  ],
  "family": null,
  "families": [
    "quartic"
  ],
  "naive_height": 101.6621040494398,
  "faltings_height": 6.540082415761474,
  "conductor": "235907306746073553639503648",
  "bad_primes": [
    "2",
    "97",
    "101",
    "197",
    "829",
    "1697"
  ],
  "discriminant": "-81987391505042662209680955414821158965248",
  "regulator": "70220731955.687175347267157344786121082832272037794175417",
  "points": [
    [
      "130896",
      "1193255208"
    ],
    [
      "6464/25",
      "6623972688/125"
    ],
    [
      "110399588/292681",
      "10134588245335164/158340421"
    ],
    [
      "3081027049/484",
      "192577290047979/10648"
    ],
    [
      "3872427264/169",
      "243456201268704/2197"
    ],
    [
      "3182337183041/407044",
      "6160748841457322367/259694072"
    ],
    [
      "4086715470309/1934881",
      "15310711056191163231/2691419471"
    ],
    [
      "61477154422052/21836929",
      "742114704260738151228/102043969217"
    ],
    [
      "75473227158436/2074164849",
      "59387364213315736974652/94463689718007"
    ]
  ],
  "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 00:01:41",
  "updated_at": "2026-11-01 00:01:41"
}