{
  "id": 2998,
  "curve_key": "265086736702321998583936070161644961:-12050400429319786947018551887520481668988886330733041",
  "ainvs": [
    "1",
    "0",
    "0",
    "-5522640347965041637165334795034270",
    "13947222719120123321051443094950791427181459592900"
  ],
  "rank_lower_bound": 8,
  "torsion": [
    6
  ],
  "naive_height": 244.6960954486401,
  "faltings_height": 18.45910645849098,
  "conductor": "61889652460509651794748110211690",
  "bad_primes": [
    "2",
    "3",
    "5",
    "11",
    "13",
    "17",
    "19",
    "23",
    "29",
    "103",
    "269",
    "1063",
    "2461489",
    "923659129"
  ],
  "discriminant": "10696002364522815410297854752573974822626731931872945518249774902061211125119163361090551144759763456000",
  "regulator": "407967375.187311270560385614486490174950472955359581581",
  "points": [
    [
      "61473486585611004/25",
      "77283175942284802749615882/125"
    ],
    [
      "157304891881646340540/49",
      "-1972409301924710939573041242210/343"
    ],
    [
      "1619279926715429100",
      "-2058381035784516976321998270"
    ],
    [
      "18332699889101287547280/73441",
      "-2370833275762318224856675155072210/19902511"
    ],
    [
      "148193842085982440122380/923521",
      "-50670582911989761687340779426191070/887503681"
    ],
    [
      "-95642736115144957730556780/1268998129",
      "-64787979088070272072835183686480794570/45205520349367"
    ],
    [
      "4662144333423838780/9",
      "-9962885593989017688498640090/27"
    ],
    [
      "2386751825253379514443461750300/5420734093009",
      "-3634726157674499948702646643486778589638094210/12620807889845925223"
    ]
  ],
  "submitter": "Natanael Wildner Fraga",
  "history": [],
  "commentary": "2-isogenous image of D-z6old678-001, rational kernel x=37773199222238940. 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 20:32:39",
  "updated_at": "2026-09-26 20:32:39"
}