{
  "id": 2968,
  "curve_key": "-424399761711351810051177633405888509279:-14222950413161855483356605882423583660250671317575955162481",
  "ainvs": [
    "1",
    "0",
    "0",
    "8841661702319829376066200695956010610",
    "16461748163381777180547691579850244647369719638524060100"
  ],
  "rank_lower_bound": 8,
  "torsion": [
    6
  ],
  "naive_height": 267.80441437371746,
  "faltings_height": 20.40682265712886,
  "conductor": "16294472568573972597369514526808180930",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "11",
    "13",
    "17",
    "23",
    "41",
    "89",
    "127",
    "167",
    "257",
    "457",
    "1459",
    "1709",
    "4603",
    "13302293"
  ],
  "discriminant": "-161303905370946951287855010660150719860320248928516361633174611351098937666689992135584958145318366186083047000000",
  "regulator": "223989878.968200193584283163279879806885307372889028653",
  "points": [
    [
      "14795452455216380138160/4489",
      "-2713669942071976282334258376064470/300763"
    ],
    [
      "11609560813282984040976/3481",
      "1871615497946263765499486642279958/205379"
    ],
    [
      "405600337069486380",
      "-4484936466204128690208299190"
    ],
    [
      "-3581659666796422474185972120/2837373289",
      "-274113493576416341270960190698974820993370/151138362985163"
    ],
    [
      "122660630226193880",
      "-4189047322128548297631402190"
    ],
    [
      "196764991383625507505790/3721",
      "-87424140908395577993918191610149890/226981"
    ],
    [
      "3570839345837157351570/24649",
      "-16302154726469824365940328082993570/3869893"
    ],
    [
      "-1497560222835004702680/1849",
      "-235444377215651261521468480377930/79507"
    ]
  ],
  "submitter": "Natanael Wildner Fraga",
  "history": [],
  "commentary": "2-isogenous image of D-z6old678-069, rational kernel x=11909531944555196831/4. 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 20:30:51",
  "updated_at": "2026-09-26 20:30:51"
}