{
  "id": 6064,
  "curve_key": "243197748034866361:-119933173000172249218728109",
  "ainvs": [
    "1",
    "0",
    "1",
    "-5066619750726383",
    "138811542402055124035218"
  ],
  "rank_lower_bound": 4,
  "torsion": [
    6
  ],
  "family": null,
  "families": [],
  "naive_height": 120.0979538557043,
  "faltings_height": 7.0416417827658515,
  "conductor": "5475946760830",
  "bad_primes": [
    "2",
    "5",
    "7",
    "13",
    "61",
    "97",
    "229",
    "4441"
  ],
  "discriminant": "119900773952893649589146000",
  "regulator": "20506.73118473402851831486605000192936309215372",
  "points": [
    [
      "657501597/16",
      "144028985/64"
    ],
    [
      "6945207363/169",
      "-45108626460/2197"
    ],
    [
      "269630228683/6561",
      "-10911840668972/531441"
    ],
    [
      "1281530796907/32041",
      "69579198982964680/5735339"
    ]
  ],
  "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=1/22204. Clearing denominators with x=X/22204^2 and y=Y/22204^3 gives [22203,-22205,-493039820,0,0]; the displayed equation is its global minimal model, with discriminant 119900773952893649589146000. 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 18:15:50",
  "updated_at": "2026-10-28 18:15:50"
}