{
  "id": 2997,
  "curve_key": "68712738716159853384804576886345401:-17981250221164721778042472029142892198537907060703149",
  "ainvs": [
    "1",
    "-1",
    "0",
    "-1431515389919996945516762018465529",
    "20811632200422132045428004921136843090461230381085"
  ],
  "rank_lower_bound": 8,
  "torsion": [
    6
  ],
  "naive_height": 240.64572802834286,
  "faltings_height": 17.846845938979364,
  "conductor": "4538897581549887969893709893633970",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "11",
    "13",
    "17",
    "19",
    "31",
    "43",
    "61",
    "67",
    "71",
    "79",
    "109",
    "181",
    "211",
    "829",
    "1479139"
  ],
  "discriminant": "635269273497117346294109213898025615539682023610261922995802423555480790026201770052632342232250000",
  "regulator": "14203200.8963730818725983609505100445962309762885784123",
  "points": [
    [
      "113265976883053686289129/4981824",
      "1471063302504282505610439820683361/11119431168"
    ],
    [
      "798577780426404663920515/27867841",
      "268109804036391292986124040751929055/147114332639"
    ],
    [
      "48180710543571511614/2809",
      "170013422398836183809263813779/148877"
    ],
    [
      "44619085686831796164654/3523129",
      "14356794668711921050552205210662611/6612913133"
    ],
    [
      "142811283164582221",
      "-52240024406178104568223473"
    ],
    [
      "262322077618318105289806/2569609",
      "-126193259236579049896317198271537947/4119083227"
    ],
    [
      "37283047612579403238247873044530645660378785885/1843711933809296536780536488464",
      "912820339366264361276541555950791892948250508416997268775749045880405/2503453909605156783619597903725490073235018688"
    ],
    [
      "122722593494811496210269924391771947931202443941/5075316903522818427734486732521",
      "-6618442584558105228251082692536504441305933193102282900299835001269812/11433909416683796838880532837342237665073447819"
    ]
  ],
  "submitter": "Natanael Wildner Fraga",
  "history": [],
  "commentary": "2-isogenous image of D-z6old678-118, rational kernel x=-168849298615036365/4. 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 20:32:36",
  "updated_at": "2026-09-26 20:32:36"
}