{
  "id": 6056,
  "curve_key": "49988830981321:-353434931876520853381",
  "ainvs": [
    "1",
    "0",
    "1",
    "-1041433978778",
    "409068121404252756"
  ],
  "rank_lower_bound": 3,
  "torsion": [
    6
  ],
  "family": null,
  "families": [],
  "naive_height": 94.62846214809001,
  "faltings_height": 4.996247798421184,
  "conductor": "587385190",
  "bad_primes": [
    "2",
    "5",
    "7",
    "17",
    "19",
    "83",
    "313"
  ],
  "discriminant": "5528655550646711000",
  "regulator": "646.07205673815363243562530742439751549878919",
  "points": [
    [
      "5494990/9",
      "763296059/27"
    ],
    [
      "71376350/121",
      "841837987/1331"
    ],
    [
      "8769500255/14884",
      "-534937610339/1815848"
    ]
  ],
  "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=-1/5321. Clearing denominators with x=X/5321^2 and y=Y/5321^3 gives [5322,5320,28307720,0,0]; the displayed equation is its global minimal model, with discriminant 5528655550646711000. 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 17:42:40",
  "updated_at": "2026-10-28 17:42:40"
}