{
  "id": 2973,
  "curve_key": "3212098836265865746576379684706439711696:-2088853328068232033402688674459259505117215070439227698631744",
  "ainvs": [
    "0",
    "1",
    "0",
    "-66918725755538869720341243431384160660",
    "2417654314893787075675250094261185544719625917049755856900"
  ],
  "rank_lower_bound": 8,
  "torsion": [
    6
  ],
  "naive_height": 277.7834416964008,
  "faltings_height": 21.206448108833012,
  "conductor": "61421217691205004364755396521722860",
  "bad_primes": [
    "2",
    "3",
    "5",
    "11",
    "13",
    "37",
    "41",
    "53",
    "101",
    "199",
    "233",
    "281",
    "359",
    "691",
    "751",
    "10259",
    "35401"
  ],
  "discriminant": "-2505883763249385080534153571503574718847513366714161766848317180726078213356877137988568622539766567323861292436000000",
  "regulator": "1384175602.25488539786959236993679688679859721838493498",
  "points": [
    [
      "23120691264139928064",
      "115021697915831925881969955330"
    ],
    [
      "104074081885249056120",
      "1059586084792223318319812312490"
    ],
    [
      "164060508251642330505/16",
      "3392342524637473083401432450925/64"
    ],
    [
      "24726012636598507651954315755/1253230801",
      "-4156540239840012642300465631774115583543150/44365623586201"
    ],
    [
      "-6862588720731736649099880/619369",
      "20673989627474542281251800210326551250/487443403"
    ],
    [
      "-9210853357251728549520/1849",
      "4075377183193271188217083945269270/79507"
    ],
    [
      "-75893620540089655383594684840/8307775609",
      "36050861757244800281469774109144787121775230/757228823433523"
    ],
    [
      "-5253817547896881059040/529",
      "-557912846129235277170069994969170/12167"
    ]
  ],
  "submitter": "Natanael Wildner Fraga",
  "history": [],
  "commentary": "2-isogenous image of D-z6old678-004, rational kernel x=7537968017313745980. Source: https://dujella.github.io/tors/z6old678.html. Independent witness points transported by the degree-2 isogeny. Witnesses saturated by exact rational halves of published and transported relations; quadratic-character independence is certified.",
  "created_at": "2026-09-26 20:31:12",
  "updated_at": "2026-09-26 20:31:12"
}