{
  "id": 5003,
  "curve_key": "8112:0",
  "ainvs": [
    "0",
    "0",
    "0",
    "-169",
    "0"
  ],
  "rank_lower_bound": 1,
  "torsion": [
    2,
    2
  ],
  "family": "congruent",
  "families": [
    "quartic",
    "congruent"
  ],
  "naive_height": 27.00329917749289,
  "faltings_height": 0.3185153430992315,
  "conductor": "5408",
  "bad_primes": [
    "2",
    "13"
  ],
  "discriminant": "308915776",
  "regulator": "5.832511736987650557951255312373859988603",
  "points": [
    [
      "-36/25",
      "1938/125"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Congruent-number curve y^2 = x^3 - n^2 x with n = 13. Found in an exhaustive ellrank scan of squarefree n <= 20000; one of the smallest n of rank 1.",
  "created_at": "2026-10-21 01:46:19",
  "updated_at": "2026-10-21 01:46:19"
}