{
  "id": 3164,
  "curve_key": "20784:2965248",
  "ainvs": [
    "0",
    "0",
    "0",
    "-433",
    "-3432"
  ],
  "rank_lower_bound": 2,
  "torsion": [
    2,
    2
  ],
  "naive_height": 29.82581621673114,
  "faltings_height": 0.35229769267731503,
  "conductor": "41440",
  "bad_primes": [
    "2",
    "5",
    "7",
    "37"
  ],
  "discriminant": "107329600",
  "regulator": "8.05665934881444914141732666545029128508279",
  "points": [
    [
      "-12",
      "6"
    ],
    [
      "149",
      "1800"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Cremona 41440b1 = LMFDB 41440.e3: https://www.lmfdb.org/EllipticCurve/Q/41440/e/3\n\nMordell–Weil rank exactly 2 (Cremona's tables). Conductor 41440, minimal discriminant 107329600, torsion ℤ/2ℤ × ℤ/2ℤ. The witnesses are the generators from Cremona's tables.",
  "created_at": "2026-09-26 20:40:20",
  "updated_at": "2026-09-26 20:40:20"
}