{
  "id": 2679,
  "curve_key": "448:-5120",
  "ainvs": [
    "0",
    "-1",
    "0",
    "-9",
    "9"
  ],
  "rank_lower_bound": 1,
  "torsion": [
    2,
    2
  ],
  "naive_height": 18.314379697244956,
  "faltings_height": -0.42622037050316613,
  "conductor": "192",
  "bad_primes": [
    "2",
    "3"
  ],
  "discriminant": "36864",
  "regulator": "0.6758018672060416661469060359761463402574",
  "points": [
    [
      "5",
      "8"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Cremona 192a2 = LMFDB 192.a2: https://www.lmfdb.org/EllipticCurve/Q/192/a/2\n\nMordell–Weil rank exactly 1 (Cremona's tables). Conductor 192, minimal discriminant 36864, torsion ℤ/2ℤ × ℤ/2ℤ. The witness is the generator from Cremona's tables.",
  "created_at": "2026-09-26 20:17:24",
  "updated_at": "2026-09-26 20:17:24"
}