{
  "id": 2933,
  "curve_key": "4209:-190377",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-88",
    "242"
  ],
  "rank_lower_bound": 3,
  "torsion": [
    2
  ],
  "naive_height": 25.034941106311713,
  "faltings_height": 0.11508891597623029,
  "conductor": "341185",
  "bad_primes": [
    "5",
    "13",
    "29",
    "181"
  ],
  "discriminant": "22177025",
  "regulator": "6.3942574871555130875014712967949363204309602",
  "points": [
    [
      "8",
      "-2"
    ],
    [
      "2",
      "7"
    ],
    [
      "23",
      "88"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Cremona 341185c1 = LMFDB 341185.c1: https://www.lmfdb.org/EllipticCurve/Q/341185/c/1\n\nMordell–Weil rank exactly 3 (Cremona's tables). Conductor 341185, minimal discriminant 22177025, torsion ℤ/2ℤ. The witnesses are the generators from Cremona's tables.",
  "created_at": "2026-09-26 20:28:58",
  "updated_at": "2026-09-26 20:28:58"
}