{
  "id": 3102,
  "curve_key": "921:4131",
  "ainvs": [
    "1",
    "-1",
    "0",
    "-19",
    "0"
  ],
  "rank_lower_bound": 1,
  "torsion": [
    2,
    2
  ],
  "naive_height": 20.47638010876592,
  "faltings_height": -0.2267350499213063,
  "conductor": "665",
  "bad_primes": [
    "5",
    "7",
    "19"
  ],
  "discriminant": "442225",
  "regulator": "2.074054851350538613018854330110751035362",
  "points": [
    [
      "16",
      "52"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Cremona 665b2 = LMFDB 665.e2: https://www.lmfdb.org/EllipticCurve/Q/665/e/2\n\nMordell–Weil rank exactly 1 (Cremona's tables). Conductor 665, minimal discriminant 442225, torsion ℤ/2ℤ × ℤ/2ℤ. The witness is the generator from Cremona's tables.",
  "created_at": "2026-09-26 20:37:24",
  "updated_at": "2026-09-26 20:37:24"
}