{
  "id": 6063,
  "curve_key": "439017595984:-290886047713997632",
  "ainvs": [
    "0",
    "1",
    "0",
    "-9146199916",
    "336670617602340"
  ],
  "rank_lower_bound": 3,
  "torsion": [
    6
  ],
  "family": null,
  "families": [],
  "naive_height": 80.42341599355929,
  "faltings_height": 3.935842112716335,
  "conductor": "134999060",
  "bad_primes": [
    "2",
    "5",
    "7",
    "29",
    "41",
    "811"
  ],
  "discriminant": "288751100782780160",
  "regulator": "317.73911305296625302603065658118689490559232",
  "points": [
    [
      "6681040/121",
      "110590/1331"
    ],
    [
      "9331336/169",
      "2198/2197"
    ],
    [
      "15958816/289",
      "11626042/4913"
    ]
  ],
  "submitter": "The Wren",
  "history": [],
  "commentary": "Obtained in my own rational-parameter search of the Kubert-Tate family for torsion 6, following the construction described by David Renshaw on curve #1192 and the t=1/6916 example submitted by Lewis Combes on curve #5970. In y^2+(1-c)*x*y-b*y=x^3-b*x^2, set c=t and b=t*(t+1), then specialize at t=1/811. Clearing denominators with x=X/811^2 and y=Y/811^3 gives [810,-812,-658532,0,0]; the displayed equation is its global minimal model, with discriminant 288751100782780160. 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/6Z.",
  "created_at": "2026-10-28 18:10:54",
  "updated_at": "2026-10-28 18:10:54"
}