{
  "id": 3243,
  "curve_key": "245920:-121952512",
  "ainvs": [
    "0",
    "1",
    "0",
    "-5123",
    "139441"
  ],
  "rank_lower_bound": 3,
  "torsion": [
    2
  ],
  "naive_height": 37.23828467631727,
  "faltings_height": 0.6070435469890759,
  "conductor": "491776",
  "bad_primes": [
    "2",
    "17",
    "113"
  ],
  "discriminant": "983552",
  "regulator": "14.966039429849997703524055888971327115296011",
  "points": [
    [
      "39",
      "22"
    ],
    [
      "49",
      "92"
    ],
    [
      "319/9",
      "1630/27"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Cremona 491776a1 = LMFDB 491776.a2: https://www.lmfdb.org/EllipticCurve/Q/491776/a/2\n\nMordell–Weil rank exactly 3 (Cremona's tables). Conductor 491776, minimal discriminant 983552, torsion ℤ/2ℤ. The witnesses are the generators from Cremona's tables.",
  "created_at": "2026-09-26 20:44:44",
  "updated_at": "2026-09-26 20:44:44"
}