{
  "id": 2795,
  "curve_key": "636048:-409855680",
  "ainvs": [
    "0",
    "0",
    "0",
    "-13251",
    "474370"
  ],
  "rank_lower_bound": 3,
  "torsion": [
    2,
    2
  ],
  "naive_height": 40.08908793351758,
  "faltings_height": 1.3473620895363203,
  "conductor": "199728",
  "bad_primes": [
    "2",
    "3",
    "19",
    "73"
  ],
  "discriminant": "51699091083264",
  "regulator": "6.2323966483756369199046582453994941097286232",
  "points": [
    [
      "113",
      "648"
    ],
    [
      "98",
      "342"
    ],
    [
      "17",
      "504"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Cremona 199728a2 = LMFDB 199728.b2: https://www.lmfdb.org/EllipticCurve/Q/199728/b/2\n\nMordell–Weil rank exactly 3 (Cremona's tables). Conductor 199728, minimal discriminant 51699091083264, torsion ℤ/2ℤ × ℤ/2ℤ. The witnesses are the generators from Cremona's tables.",
  "created_at": "2026-09-26 20:22:46",
  "updated_at": "2026-09-26 20:22:46"
}