{
  "id": 870,
  "curve_key": "0:-216",
  "ainvs": [
    "0",
    "0",
    "1",
    "0",
    "0"
  ],
  "rank_lower_bound": 0,
  "torsion": [
    3
  ],
  "naive_height": 10.75055681536833,
  "faltings_height": -1.0464643562610105,
  "conductor": "27",
  "bad_primes": [
    "3"
  ],
  "discriminant": "-27",
  "regulator": "1",
  "points": [],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "LMFDB 27.a4 (Cremona 27a3): https://www.lmfdb.org/EllipticCurve/Q/27/a/4\r\n\r\nMordell–Weil rank exactly 0 per LMFDB. Conductor 27, minimal discriminant -27, torsion ℤ/3ℤ.\r\n\r\nSubmitted as one of the ten smallest rank-0 curves in LMFDB (as of 2026-09-23; places in LMFDB sort order, ties broken by label): #5 by Faltings height, #10 by |minimal discriminant|.\r\n\r\nFirst CM curve.  The 3-isogenous j=0 curve [0,0,1,0,-7] also has rational 3-torsion (generated by [3,4]) and is isomorphic with the Fermat cubic curve X³ + Y³ = Z³.",
  "created_at": "2026-09-23 14:40:25",
  "updated_at": "2026-09-23 14:40:25"
}