{
  "id": 1012,
  "curve_key": "-128:-1144",
  "ainvs": [
    "0",
    "1",
    "1",
    "3",
    "2"
  ],
  "rank_lower_bound": 1,
  "torsion": [
    3
  ],
  "naive_height": 14.556090791758852,
  "faltings_height": -0.6806234233412365,
  "conductor": "219",
  "bad_primes": [
    "3",
    "73"
  ],
  "discriminant": "-1971",
  "regulator": "1.256236863853751532911588770361478097847",
  "points": [
    [
      "2",
      "4"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Cremona 219b1 = LMFDB 219.b2: https://www.lmfdb.org/EllipticCurve/Q/219/b/2\n\nMordell–Weil rank exactly 1 (Cremona's tables). Conductor 219, minimal discriminant -1971, torsion ℤ/3ℤ. The witness is the generator from Cremona's tables.",
  "created_at": "2026-09-23 15:14:23",
  "updated_at": "2026-09-23 15:14:23"
}