{
  "id": 3228,
  "curve_key": "4144:-266752",
  "ainvs": [
    "0",
    "1",
    "0",
    "-86",
    "280"
  ],
  "rank_lower_bound": 2,
  "torsion": [
    2
  ],
  "naive_height": 24.988250351817957,
  "faltings_height": -0.24294117439023646,
  "conductor": "2080",
  "bad_primes": [
    "2",
    "5",
    "13"
  ],
  "discriminant": "4160",
  "regulator": "1.19405006443153500984712892878588038131964",
  "points": [
    [
      "6",
      "4"
    ],
    [
      "1",
      "14"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Cremona 2080d1 = LMFDB 2080.a1: https://www.lmfdb.org/EllipticCurve/Q/2080/a/1\n\nMordell–Weil rank exactly 2 (Cremona's tables). Conductor 2080, minimal discriminant 4160, torsion ℤ/2ℤ. The witnesses are the generators from Cremona's tables.",
  "created_at": "2026-09-26 20:43:36",
  "updated_at": "2026-09-26 20:43:36"
}