{
  "id": 2686,
  "curve_key": "81:135",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-2",
    "0"
  ],
  "rank_lower_bound": 1,
  "torsion": [
    2
  ],
  "naive_height": 13.183347464017316,
  "faltings_height": -0.8352641639743923,
  "conductor": "99",
  "bad_primes": [
    "3",
    "11"
  ],
  "discriminant": "297",
  "regulator": "0.3025713845614949991761430348115623877720",
  "points": [
    [
      "0",
      "0"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Cremona 99a1 = LMFDB 99.a2: https://www.lmfdb.org/EllipticCurve/Q/99/a/2\n\nMordell–Weil rank exactly 1 (Cremona's tables). Conductor 99, minimal discriminant 297, torsion ℤ/2ℤ. The witness is the generator from Cremona's tables.",
  "created_at": "2026-09-26 20:17:43",
  "updated_at": "2026-09-26 20:17:43"
}