{
  "id": 2882,
  "curve_key": "8368:-385280",
  "ainvs": [
    "0",
    "-1",
    "0",
    "-174",
    "504"
  ],
  "rank_lower_bound": 2,
  "torsion": [
    2,
    2
  ],
  "naive_height": 27.09651055891411,
  "faltings_height": 0.30873684844576066,
  "conductor": "21216",
  "bad_primes": [
    "2",
    "3",
    "13",
    "17"
  ],
  "discriminant": "253191744",
  "regulator": "3.75248831544398869761568820361457024176983",
  "points": [
    [
      "-6",
      "36"
    ],
    [
      "-13",
      "20"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Cremona 21216l1 = LMFDB 21216.d3: https://www.lmfdb.org/EllipticCurve/Q/21216/d/3\n\nMordell–Weil rank exactly 2 (Cremona's tables). Conductor 21216, minimal discriminant 253191744, torsion ℤ/2ℤ × ℤ/2ℤ. The witnesses are the generators from Cremona's tables.",
  "created_at": "2026-09-26 20:26:41",
  "updated_at": "2026-09-26 20:26:41"
}