{
  "id": 972,
  "curve_key": "-47:199",
  "ainvs": [
    "1",
    "1",
    "1",
    "1",
    "0"
  ],
  "rank_lower_bound": 1,
  "torsion": [],
  "naive_height": 11.550442805130176,
  "faltings_height": -0.9438672968611436,
  "conductor": "83",
  "bad_primes": [
    "83"
  ],
  "discriminant": "-83",
  "regulator": "0.1772922941147684434576006159994158304290",
  "points": [
    [
      "0",
      "0"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Cremona 83a1 = LMFDB 83.a1: https://www.lmfdb.org/EllipticCurve/Q/83/a/1\n\nMordell–Weil rank exactly 1 (Cremona's tables). Conductor 83, minimal discriminant -83, trivial torsion. The witness is the generator from Cremona's tables.",
  "created_at": "2026-09-23 15:08:58",
  "updated_at": "2026-09-23 15:08:58"
}