{
  "id": 6053,
  "curve_key": "83955547732905:-769261679973281751957",
  "ainvs": [
    "1",
    "-1",
    "0",
    "-1749073911102",
    "890349603904220544"
  ],
  "rank_lower_bound": 3,
  "torsion": [
    6
  ],
  "family": null,
  "families": [],
  "naive_height": 96.18392574313076,
  "faltings_height": 5.118370056179455,
  "conductor": "64192086",
  "bad_primes": [
    "2",
    "3",
    "7",
    "673",
    "757"
  ],
  "discriminant": "12031633142133814692",
  "regulator": "1171.6320458645895638286154161774722364641136",
  "points": [
    [
      "37414350/49",
      "-129607416/343"
    ],
    [
      "1039341099/100",
      "33242942744337/1000"
    ],
    [
      "15614054119/20449",
      "-1116330431373/2924207"
    ]
  ],
  "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=-1/6057. Clearing denominators with x=X/6057^2 and y=Y/6057^3 gives [6058,6056,36681192,0,0]; the displayed equation is its global minimal model, with discriminant 12031633142133814692. 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 17:36:07",
  "updated_at": "2026-10-28 17:36:07"
}