{
  "id": 2172,
  "curve_key": "500555536:-7882980410816",
  "ainvs": [
    "0",
    "-1",
    "0",
    "-10428240",
    "9127296000"
  ],
  "rank_lower_bound": 3,
  "torsion": [
    4,
    2
  ],
  "naive_height": 60.09368733480837,
  "faltings_height": 3.0355865684102645,
  "conductor": "14769840",
  "bad_primes": [
    "2",
    "3",
    "5",
    "19",
    "41",
    "79"
  ],
  "discriminant": "36617902166962852761600",
  "regulator": "58.343701545475895555207749666826711159366458",
  "points": [
    [
      "-2320",
      "144320"
    ],
    [
      "-2415",
      "142200"
    ],
    [
      "-2688",
      "133152"
    ]
  ],
  "submitter": "Steps Unbounded",
  "history": [],
  "commentary": "Height-ordered search of the universal family for torsion Z/2Z x Z/4Z (Tate/Kubert normal form, parameter t=15/19), smallest curves first; rank lower bound from PARI ellrank: 3 independent points.",
  "created_at": "2026-09-26 14:55:43",
  "updated_at": "2026-09-26 14:55:43"
}