{
  "id": 6060,
  "curve_key": "9219915104181529:-885299834873060718730237",
  "ainvs": [
    "1",
    "0",
    "1",
    "-192081564670449",
    "1024652570651838035252"
  ],
  "rank_lower_bound": 4,
  "torsion": [
    6
  ],
  "family": null,
  "families": [],
  "naive_height": 110.28042667394212,
  "faltings_height": 6.26615183369804,
  "conductor": "470499847330",
  "bad_primes": [
    "2",
    "5",
    "31",
    "37",
    "53",
    "79",
    "97",
    "101"
  ],
  "discriminant": "884757952795108520036240",
  "regulator": "9072.195226108893241464293083463335783182775279",
  "points": [
    [
      "28799115/4",
      "30873791879/8"
    ],
    [
      "8011303",
      "43058554"
    ],
    [
      "6728910459/841",
      "-23913016570/24389"
    ],
    [
      "1700528887418/212521",
      "-391552286934041/97972181"
    ]
  ],
  "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=1/9796. Clearing denominators with x=X/9796^2 and y=Y/9796^3 gives [9795,-9797,-95971412,0,0]; the displayed equation is its global minimal model, with discriminant 884757952795108520036240. All four listed rational points were checked exactly and returned as an independent set, with rank bounds [4,4]. The rational torsion subgroup is Z/6Z.",
  "created_at": "2026-10-28 18:01:38",
  "updated_at": "2026-10-28 18:01:38"
}