{
  "id": 646,
  "curve_key": "49:-73",
  "ainvs": [
    "1",
    "0",
    "0",
    "-1",
    "0"
  ],
  "rank_lower_bound": 1,
  "torsion": [
    2
  ],
  "naive_height": 11.675460894331879,
  "faltings_height": -0.9616152372921617,
  "conductor": "65",
  "bad_primes": [
    "5",
    "13"
  ],
  "discriminant": "65",
  "regulator": "0.3755140986612663218044728768245783176621",
  "points": [
    [
      "1",
      "0"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Cremona 65a1 = LMFDB 65.a1. Curve data, generator (1, 0) and rank from the LMFDB: https://www.lmfdb.org/EllipticCurve/Q/65/a/1\r\n\r\nThis curve and the 2-isogenous 65a2 are the curves of smallest conductor that have both positive rank and nontrivial torsion.",
  "created_at": "2026-09-08 19:03:16",
  "updated_at": "2026-09-08 19:03:16"
}