{
  "id": 3003,
  "curve_key": "-7167:-2681505",
  "ainvs": [
    "1",
    "-1",
    "1",
    "149",
    "3066"
  ],
  "rank_lower_bound": 2,
  "torsion": [
    4
  ],
  "naive_height": 29.60377752406277,
  "faltings_height": 0.5308208938812639,
  "conductor": "51359",
  "bad_primes": [
    "7",
    "11",
    "23",
    "29"
  ],
  "discriminant": "-4374194671",
  "regulator": "10.0445113880872572176935395003776252499751",
  "points": [
    [
      "-2",
      "53"
    ],
    [
      "1116",
      "36713"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Cremona 51359g1 = LMFDB 51359.e4: https://www.lmfdb.org/EllipticCurve/Q/51359/e/4\n\nMordell–Weil rank exactly 2 (Cremona's tables). Conductor 51359, minimal discriminant -4374194671, torsion ℤ/4ℤ. The witnesses are the generators from Cremona's tables.",
  "created_at": "2026-09-26 20:32:51",
  "updated_at": "2026-09-26 20:32:51"
}