{
  "checked_at": "2026-10-04",
  "curves": [
    {
      "id": "E1",
      "equation": "y^2 = x^3 - 2",
      "ainvs": [0, 0, 0, 0, -2],
      "lmfdb_label": "1728.o3",
      "conductor": 1728,
      "minimal_discriminant": -1728,
      "bad_primes": [2, 3],
      "algebraic_rank": 1,
      "analytic_rank": 1,
      "torsion_structure": [],
      "generators": [[3, 5, 1]],
      "root_number": -1,
      "source": "https://www.lmfdb.org/EllipticCurve/Q/data/1728.o3"
    },
    {
      "id": "E0",
      "equation": "y^2 = x^3 - x",
      "ainvs": [0, 0, 0, -1, 0],
      "lmfdb_label": "32.a3",
      "conductor": 32,
      "minimal_discriminant": 64,
      "bad_primes": [2],
      "algebraic_rank": 0,
      "analytic_rank": 0,
      "torsion_structure": [2, 2],
      "torsion_points": ["O", [0, 0], [1, 0], [-1, 0]],
      "root_number": 1,
      "source": "https://www.lmfdb.org/EllipticCurve/Q/32/a/3"
    }
  ],
  "notes": [
    "LMFDB の ainvs は一般 Weierstrass 形 [a1,a2,a3,a4,a6] の係数である。",
    "E1 の生成元 (3,5) と rank bounds [1,1] は LMFDB ec_mwbsd データで確認した。",
    "E0 の torsion 点は方程式への代入と 2-torsion 条件 y=0 から直接確認できる。"
  ]
}