{
  "id": 3291,
  "curve_key": "19921:807751",
  "ainvs": [
    "1",
    "1",
    "1",
    "-415",
    "-1108"
  ],
  "rank_lower_bound": 2,
  "torsion": [
    4
  ],
  "naive_height": 29.69858919204535,
  "faltings_height": 0.5379077652839276,
  "conductor": "46965",
  "bad_primes": [
    "3",
    "5",
    "31",
    "101"
  ],
  "discriminant": "4197402945",
  "regulator": "9.80735191701645501408687024646217732410698",
  "points": [
    [
      "-4",
      "24"
    ],
    [
      "46",
      "259"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Cremona 46965b1 = LMFDB 46965.a4: https://www.lmfdb.org/EllipticCurve/Q/46965/a/4\n\nMordell–Weil rank exactly 2 (Cremona's tables). Conductor 46965, minimal discriminant 4197402945, torsion ℤ/4ℤ. The witnesses are the generators from Cremona's tables.",
  "created_at": "2026-09-26 20:52:34",
  "updated_at": "2026-09-26 20:52:34"
}