{
  "id": 6365,
  "curve_key": "6709651094419967539489:-549604026523229203134917797887793",
  "ainvs": [
    "1",
    "0",
    "0",
    "-139784397800415990406",
    "636115771427273988996860933540"
  ],
  "rank_lower_bound": 6,
  "torsion": [
    6
  ],
  "family": null,
  "families": [],
  "naive_height": 150.77350171396694,
  "faltings_height": 9.879965070419505,
  "conductor": "311673756083482470",
  "bad_primes": [
    "2",
    "3",
    "5",
    "17",
    "23",
    "29",
    "37",
    "113",
    "379",
    "578209"
  ],
  "discriminant": "440345528228360941904918005448472603885279998197440",
  "regulator": "155072.36902002506896414742400295276566105142404774",
  "points": [
    [
      "11886019406",
      "808611584554724"
    ],
    [
      "334476283514/49",
      "-904855875160/343"
    ],
    [
      "9370964785310/1369",
      "138181371894995540/50653"
    ],
    [
      "12623202223034/1849",
      "11157652019956280/79507"
    ],
    [
      "47558053777046/6889",
      "6346324969063843450/571787"
    ],
    [
      "209581048016936/30625",
      "13350793717043141666/5359375"
    ]
  ],
  "submitter": "The Wren",
  "history": [],
  "commentary": "Z/6-torsion Kubert–Tate family curve from my rational-parameter search of y^2+(1-c)xy-by=x^3-bx^2, with b=c(c+1), following the construction of David Renshaw on curve#1192 and the c=1/6916 example of Lewis Combes on curve#5970.\r\n\r\nHere c=-1921/578209. An integral model is [290065,276762312,80013229829604,0,0], which reduces to the displayed global minimal model.\r\n\r\nThe completed rank-6 batch identified this specialization. A separate rank computation on the minimal model returned bounds [6,6] and the six displayed rational points. All six points were checked exactly on this model. The minimal discriminant was fully factored and every listed bad-reduction prime proved prime.",
  "created_at": "2026-11-01 14:27:16",
  "updated_at": "2026-11-01 14:27:16"
}