{
  "id": 2664,
  "curve_key": "112:424",
  "ainvs": [
    "0",
    "-1",
    "1",
    "-2",
    "0"
  ],
  "rank_lower_bound": 2,
  "torsion": [],
  "naive_height": 14.155496613885283,
  "faltings_height": -0.760408353602041,
  "conductor": "709",
  "bad_primes": [
    "709"
  ],
  "discriminant": "709",
  "regulator": "0.258683172736406669969244323486181605005230",
  "points": [
    [
      "0",
      "0"
    ],
    [
      "-1",
      "0"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Cremona 709a1 = LMFDB 709.a1: https://www.lmfdb.org/EllipticCurve/Q/709/a/1\n\nMordell–Weil rank exactly 2 (Cremona's tables). Conductor 709, minimal discriminant 709, torsion trivial. The witnesses are the generators from Cremona's tables.",
  "created_at": "2026-09-26 20:16:36",
  "updated_at": "2026-09-26 20:16:36"
}