{
  "id": 2824,
  "curve_key": "1141312:-1218970936",
  "ainvs": [
    "0",
    "1",
    "1",
    "-23777",
    "1402920"
  ],
  "rank_lower_bound": 3,
  "torsion": [
    3
  ],
  "naive_height": 41.84306710743655,
  "faltings_height": 1.2134835804317947,
  "conductor": "413231",
  "bad_primes": [
    "7",
    "13",
    "19",
    "239"
  ],
  "discriminant": "448479191069",
  "regulator": "19.343358243019390928932869330677215701986880",
  "points": [
    [
      "90",
      "9"
    ],
    [
      "166",
      "1434"
    ],
    [
      "-176",
      "408"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Cremona 413231b1 = LMFDB 413231.b2: https://www.lmfdb.org/EllipticCurve/Q/413231/b/2\n\nMordell–Weil rank exactly 3 (Cremona's tables). Conductor 413231, minimal discriminant 448479191069, torsion ℤ/3ℤ. The witnesses are the generators from Cremona's tables.",
  "created_at": "2026-09-26 20:24:04",
  "updated_at": "2026-09-26 20:24:04"
}