{
  "id": 6050,
  "curve_key": "126904816:-1535507177048",
  "ainvs": [
    "0",
    "-1",
    "1",
    "-2643850",
    "1778088664"
  ],
  "rank_lower_bound": 3,
  "torsion": [
    5
  ],
  "family": null,
  "families": [],
  "naive_height": 56.11976370179033,
  "faltings_height": 2.6310027937818328,
  "conductor": "32223851",
  "bad_primes": [
    "11",
    "23",
    "67",
    "1901"
  ],
  "discriminant": "-181713835635778551611",
  "regulator": "135.89984832083987833183813776695432954218293",
  "points": [
    [
      "1028",
      "12063"
    ],
    [
      "4478",
      "282336"
    ],
    [
      "275542/49",
      "139199129/343"
    ]
  ],
  "submitter": "The Wren",
  "history": [],
  "commentary": "Obtained in my own rational-parameter search of the Tate X_1(5) family, following the construction described by David Renshaw on curves #1185 and #1186. The family is y^2+(1-t)*x*y-t*y=x^3-t*x^2, specialized at t=-67/23. Clearing denominators with x=X/23^2 and y=Y/23^3 gives [90,1541,35443,0,0]; the displayed equation is its global minimal model, with discriminant -181713835635778551611. All three listed rational points were checked exactly and returned as an independent set, with rank bounds [3,3]. The rational torsion subgroup is Z/5Z.",
  "created_at": "2026-10-28 17:10:01",
  "updated_at": "2026-10-28 17:10:01"
}