{
  "id": 2953,
  "curve_key": "6712843215121735116719986129064933045041:4151835319805574833672042461766946050121920405909785245911",
  "ainvs": [
    "1",
    "0",
    "0",
    "-139850900315036148264999711022186105105",
    "-4805364953478674587663550097742533246761457869799982423"
  ],
  "rank_lower_bound": 8,
  "torsion": [
    6
  ],
  "naive_height": 275.1145236481201,
  "faltings_height": 20.994447254037294,
  "conductor": "26841118146961827757736036579027307510",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "11",
    "13",
    "17",
    "29",
    "53",
    "59",
    "61",
    "83",
    "101",
    "433",
    "563",
    "2339",
    "4643",
    "15107",
    "28349"
  ],
  "discriminant": "175045529873088486037810052100954891106645675554441769591839684634800691243559797760516675582044489080371127779328000",
  "regulator": "331114045.295381771050493409626337075323229645827348652",
  "points": [
    [
      "20885062392873283082",
      "-78639482173097025838962331045"
    ],
    [
      "641881605098000176814",
      "-16259555594520457935446906247937"
    ],
    [
      "12261725728017976330661594/229441",
      "-41871448477684738888946197333663722203/109902239"
    ],
    [
      "24078725825364590702",
      "-102899213335006607875627535425"
    ],
    [
      "4306755891709038836201/64",
      "278233791598561023221555072818591/512"
    ],
    [
      "-495169472677836543536049814/44849809",
      "4177442617039521213407841635279874630659/300359170873"
    ],
    [
      "-111418105141036281080584/10609",
      "19101767500565451739638500876303411/1092727"
    ],
    [
      "-89786138354773293960688/271441",
      "910136268211453289839063865170312405/141420761"
    ]
  ],
  "submitter": "Natanael Wildner Fraga",
  "history": [],
  "commentary": "2-isogenous image of D-z6old678-033, rational kernel x=274887355699892655/4. 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 20:29:54",
  "updated_at": "2026-09-26 20:29:54"
}