{
  "id": 993,
  "curve_key": "-32:424",
  "ainvs": [
    "0",
    "-1",
    "1",
    "1",
    "-1"
  ],
  "rank_lower_bound": 1,
  "torsion": [],
  "naive_height": 12.099466910463915,
  "faltings_height": -0.9161594414958077,
  "conductor": "123",
  "bad_primes": [
    "3",
    "41"
  ],
  "discriminant": "-123",
  "regulator": "0.2919145287294710320057605948504386772462",
  "points": [
    [
      "1",
      "0"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Cremona 123b1 = LMFDB 123.b1: https://www.lmfdb.org/EllipticCurve/Q/123/b/1\n\nMordell–Weil rank exactly 1 (Cremona's tables). Conductor 123, minimal discriminant -123, trivial torsion. The witness is the generator from Cremona's tables.",
  "created_at": "2026-09-23 15:12:12",
  "updated_at": "2026-09-23 15:12:12"
}