{
  "id": 6055,
  "curve_key": "514658496121:-369214475673310669",
  "ainvs": [
    "1",
    "0",
    "1",
    "-10722052003",
    "427330675561998"
  ],
  "rank_lower_bound": 3,
  "torsion": [
    6
  ],
  "family": null,
  "families": [],
  "naive_height": 80.90030820997333,
  "faltings_height": 3.972631087413429,
  "conductor": "23397790",
  "bad_primes": [
    "2",
    "5",
    "13",
    "211",
    "853"
  ],
  "discriminant": "366609784112234000",
  "regulator": "137.49640939445145744108821780735070340604142",
  "points": [
    [
      "-12041",
      "23557860"
    ],
    [
      "50277499/841",
      "-728966640/24389"
    ],
    [
      "100495214/1681",
      "-2059972140/68921"
    ]
  ],
  "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=1/844. Clearing denominators with x=X/844^2 and y=Y/844^3 gives [843,-845,-713180,0,0]; the displayed equation is its global minimal model, with discriminant 366609784112234000. 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 17:40:22",
  "updated_at": "2026-10-28 17:40:22"
}