{
  "id": 2964,
  "curve_key": "1987007812314986049757864492485552620121:-88488446226094735601643220612377363038922009055292632543581",
  "ainvs": [
    "1",
    "-1",
    "0",
    "-41395996089895542703288843593449012919",
    "102417183132054092141880504360873611415315276934580577933"
  ],
  "rank_lower_bound": 8,
  "torsion": [
    6
  ],
  "naive_height": 271.46234556613007,
  "faltings_height": 20.393421633485996,
  "conductor": "77775552131973229100114648668568825370",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "13",
    "19",
    "29",
    "31",
    "101",
    "167",
    "229",
    "397",
    "431",
    "1613",
    "7907",
    "142789",
    "461917"
  ],
  "discriminant": "8622234302599899825576507504372051098109700262434430191379942000308011224963304671601542617832701513342603626000",
  "regulator": "5230589820.10840507419805257549556834099888755662275828",
  "points": [
    [
      "19353325195586704474",
      "80932552835479910330648974925"
    ],
    [
      "75886220115783959346351381199/19647348561",
      "1063534845585928042467239571306011635962610/2753949200446809"
    ],
    [
      "6807626962769186263824233976357409/1601064496996416",
      "115930180567897533960158052532520184369967592228989/64063880441924680318464"
    ],
    [
      "24524543716861722824038866836514/26503747016041",
      "1099256988375361052245267746336211557611292599955/136446033809294939339"
    ],
    [
      "55550368404099364187822619797242/9608208283521",
      "-223549235783061855562373144143207202523581778221/29782668906721162431"
    ],
    [
      "7151108077874057906230888882856854/2103384591254241",
      "95305741701978525771447628047661672453179141914555/96466835495162732337711"
    ],
    [
      "119018464818001018571206/24649",
      "-15043457025823494530073509781823335/3869893"
    ],
    [
      "1050662633415068877306381100287/135511598161",
      "-784853584611500388198108086141779691450529879/49884394003429159"
    ]
  ],
  "submitter": "Natanael Wildner Fraga",
  "history": [],
  "commentary": "2-isogenous image of D-z6old678-016, rational kernel x=-30462022090411145901/4. 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 20:30:36",
  "updated_at": "2026-09-26 20:30:36"
}