{
  "id": 6313,
  "curve_key": "-329594808287520:0",
  "ainvs": [
    "0",
    "0",
    "0",
    "6866558505990",
    "0"
  ],
  "rank_lower_bound": 9,
  "torsion": [
    2
  ],
  "family": null,
  "families": [
    "quartic"
  ],
  "naive_height": 100.28665548685485,
  "faltings_height": 6.425461702212729,
  "conductor": "335286227315083526157369600",
  "bad_primes": [
    "2",
    "3",
    "5",
    "31",
    "173",
    "1733",
    "8209"
  ],
  "discriminant": "-20720362465005151197268879975838195136000",
  "regulator": "4981545650.0087368747120099407700589480242335071039927391",
  "points": [
    [
      "10",
      "8286470"
    ],
    [
      "6162250",
      "16622712550"
    ],
    [
      "8209/4",
      "949674583/8"
    ],
    [
      "1610361/4",
      "13457276739/8"
    ],
    [
      "53859249/100",
      "1963291308057/1000"
    ],
    [
      "213662785/1156",
      "44388400163345/39304"
    ],
    [
      "368502010/49",
      "7491035359970/343"
    ],
    [
      "44680675801/4356",
      "9747837429912179/287496"
    ],
    [
      "60847040718751/158785201",
      "3280153470782164768421/2000852317801"
    ]
  ],
  "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 00:01:03",
  "updated_at": "2026-11-01 00:01:03"
}