{
  "id": 3238,
  "curve_key": "-4592:-141760",
  "ainvs": [
    "0",
    "1",
    "0",
    "96",
    "196"
  ],
  "rank_lower_bound": 3,
  "torsion": [
    2
  ],
  "naive_height": 25.296212813998206,
  "faltings_height": 0.19103139844258382,
  "conductor": "81328",
  "bad_primes": [
    "2",
    "13",
    "17",
    "23"
  ],
  "discriminant": "-67664896",
  "regulator": "3.5440189536549500648617999498029826487372104",
  "points": [
    [
      "2",
      "20"
    ],
    [
      "6",
      "32"
    ],
    [
      "-1",
      "10"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Cremona 81328c1 = LMFDB 81328.b2: https://www.lmfdb.org/EllipticCurve/Q/81328/b/2\n\nMordell–Weil rank exactly 3 (Cremona's tables). Conductor 81328, minimal discriminant -67664896, torsion ℤ/2ℤ. The witnesses are the generators from Cremona's tables.",
  "created_at": "2026-09-26 20:44:05",
  "updated_at": "2026-09-26 20:44:05"
}