{
  "id": 2817,
  "curve_key": "552832:-411013504",
  "ainvs": [
    "0",
    "1",
    "0",
    "-11517",
    "471871"
  ],
  "rank_lower_bound": 3,
  "torsion": [
    3
  ],
  "naive_height": 39.668428310640955,
  "faltings_height": 0.9934365603490815,
  "conductor": "447764",
  "bad_primes": [
    "2",
    "23",
    "31",
    "157"
  ],
  "discriminant": "15159497984",
  "regulator": "12.054093319166366349113431520062521951543033",
  "points": [
    [
      "61",
      "6"
    ],
    [
      "57",
      "62"
    ],
    [
      "-79",
      "946"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Cremona 447764a1 = LMFDB 447764.a2: https://www.lmfdb.org/EllipticCurve/Q/447764/a/2\n\nMordell–Weil rank exactly 3 (Cremona's tables). Conductor 447764, minimal discriminant 15159497984, torsion ℤ/3ℤ. The witnesses are the generators from Cremona's tables.",
  "created_at": "2026-09-26 20:23:45",
  "updated_at": "2026-09-26 20:23:45"
}