{
  "id": 3224,
  "curve_key": "409:-13373",
  "ainvs": [
    "1",
    "1",
    "0",
    "-8",
    "12"
  ],
  "rank_lower_bound": 2,
  "torsion": [
    2
  ],
  "naive_height": 19.001986055745594,
  "faltings_height": -0.3814194379912499,
  "conductor": "2130",
  "bad_primes": [
    "2",
    "3",
    "5",
    "71"
  ],
  "discriminant": "-63900",
  "regulator": "0.342631423899141355230254981426106780108122",
  "points": [
    [
      "1",
      "2"
    ],
    [
      "6",
      "12"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Cremona 2130b1 = LMFDB 2130.a2: https://www.lmfdb.org/EllipticCurve/Q/2130/a/2\n\nMordell–Weil rank exactly 2 (Cremona's tables). Conductor 2130, minimal discriminant -63900, torsion ℤ/2ℤ. The witnesses are the generators from Cremona's tables.",
  "created_at": "2026-09-26 20:43:26",
  "updated_at": "2026-09-26 20:43:26"
}