{
  "id": 2985,
  "curve_key": "927932486038908119202076605886078441:-1861964856477352606619833905442564303031434495657439989",
  "ainvs": [
    "1",
    "0",
    "1",
    "-19331926792477252483376595955959968",
    "2155051917219158107902702750444678384523518040310558"
  ],
  "rank_lower_bound": 8,
  "torsion": [
    6
  ],
  "naive_height": 249.92245465256582,
  "faltings_height": 18.87346936043784,
  "conductor": "17783956777141537113166960878108870",
  "bad_primes": [
    "2",
    "3",
    "5",
    "11",
    "13",
    "17",
    "19",
    "43",
    "71",
    "109",
    "137",
    "211",
    "239",
    "269",
    "1033",
    "21673",
    "926921"
  ],
  "discriminant": "-1543928696699302777937944795258375620155932770150965132064118539683536438493624956450964839805577597750000",
  "regulator": "504250417.262729497188993199609760578839668691604203383",
  "points": [
    [
      "72829699846154994",
      "33666218186691143060657590"
    ],
    [
      "133043793207834069",
      "44022957423816284924663515"
    ],
    [
      "28789059517751423781/529",
      "432597663752699893798097587625/12167"
    ],
    [
      "-219382298581854778359/6889",
      "-29921389889588392646510698283995/571787"
    ],
    [
      "21202357770438572885586/69169",
      "2878099158326057116855894352397530/18191447"
    ],
    [
      "6683822577532811601/49",
      "15552920263257737810862678265/343"
    ],
    [
      "-547957753729489843382557926/3304525225",
      "-5376972297063063318376455784024662735877/189960632559125"
    ],
    [
      "2600433823957866438609045726/37819747729",
      "-249513820142902149683328229868315751743045/7354919800101817"
    ]
  ],
  "submitter": "Natanael Wildner Fraga",
  "history": [],
  "commentary": "2-isogenous image of D-z6old678-067, rational kernel x=1419372898239240055/4. 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 20:31:53",
  "updated_at": "2026-09-26 20:31:53"
}