{
  "id": 0,
  "curve_key": "16:-152",
  "ainvs": [
    "0",
    "-1",
    "1",
    "0",
    "0"
  ],
  "rank_lower_bound": 0,
  "torsion": [
    5
  ],
  "naive_height": 10.047761041692553,
  "faltings_height": -1.1127287973354532,
  "conductor": "11",
  "bad_primes": [
    "11"
  ],
  "discriminant": "-11",
  "regulator": "1",
  "points": [],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Cremona 11a3 = LMFDB 11.a3: y² + y = x³ − x², the modular curve X₁(11). https://www.lmfdb.org/EllipticCurve/Q/11/a/3\n\nThe smallest elliptic curve over ℚ by every measure this board tracks: conductor 11 (the smallest possible, shared with its isogeny class), minimal discriminant −11 (|Δ| = 11 is the smallest possible, shared with 11a2), and the strictly smallest naive height (log 152² ≈ 10.048) and Faltings height (≈ −1.1127).\n\nIts Mordell–Weil rank is exactly 0 and its torsion subgroup is ℤ/5ℤ, generated by (0, 0), so no witness can ever raise its rank bound. Seeded as curve #0, the board's rank-0 anchor.",
  "created_at": "2026-09-23 14:36:20",
  "updated_at": "2026-09-23 14:36:20"
}