{
  "id": 2992,
  "curve_key": "5209065460520358700190781154097235601:-10516630082857119936801444985289863761280626076180417849",
  "ainvs": [
    "1",
    "0",
    "0",
    "-108522197094174139587307940710359075",
    "12172025558862407325217415271422830684021359195641857"
  ],
  "rank_lower_bound": 8,
  "torsion": [
    6
  ],
  "naive_height": 253.6303914398433,
  "faltings_height": 19.11690181145301,
  "conductor": "3236200724342912634084121282477610",
  "bad_primes": [
    "2",
    "5",
    "7",
    "19",
    "31",
    "37",
    "59",
    "127",
    "151",
    "163",
    "179",
    "181",
    "827",
    "1039",
    "7621",
    "54217"
  ],
  "discriminant": "17792340654479391758344901961217543732181119755783703740971135244704202046091794561714056709231614464000000",
  "regulator": "235201498.679013008874681970223454604234327074754734719",
  "points": [
    [
      "162141664482702488528084/130321",
      "63172696208493324537524820236503563/47045881"
    ],
    [
      "127683928492640830856/289",
      "1103696413388751454870644158049/4913"
    ],
    [
      "417061422988397414206/225",
      "8389940293661477614024704920359/3375"
    ],
    [
      "-292976632583584306",
      "137181464463385666783629513"
    ],
    [
      "294929844651866317754686/727609",
      "-115750296510221044061813619248611979/620650477"
    ],
    [
      "6005527231598362898207806/5212089",
      "14163899325082280321706665421981304651/11899199187"
    ],
    [
      "7563938823219868567143601104526/6572337213649",
      "-20018870208302140085973844946144088131178709631/16849218304131754393"
    ],
    [
      "72500225983722135520626895262/708070077961",
      "27522477908534608462653100692012795888936365/595819020431764709"
    ]
  ],
  "submitter": "Natanael Wildner Fraga",
  "history": [],
  "commentary": "2-isogenous image of D-z6old678-053, rational kernel x=-67332390755112274. 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 20:32:21",
  "updated_at": "2026-09-26 20:32:21"
}