{
  "id": 2948,
  "curve_key": "715780035594928627505556174891120303604321:-603997468136999574147833652095880743359558255196012159266564369",
  "ainvs": [
    "1",
    "0",
    "0",
    "-14912084074894346406365753643565006325090",
    "699071143677082840447638868104731887058217501080642915290692"
  ],
  "rank_lower_bound": 8,
  "torsion": [
    6
  ],
  "naive_height": 289.12257460101796,
  "faltings_height": 21.903245988982277,
  "conductor": "194702283408067369042637179695724289610",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "11",
    "13",
    "17",
    "19",
    "23",
    "29",
    "31",
    "107",
    "109",
    "317",
    "42569",
    "75217",
    "99881",
    "821039"
  ],
  "discriminant": "1105648256292700046714660322063142704096721866323889318170417461479210467838014509010249740571649373913818824000000000000",
  "regulator": "259382365.825469450149368411565441159771860512261759624",
  "points": [
    [
      "514054065361445680324",
      "11352226819454827981453856409838"
    ],
    [
      "705260194295013500416/9",
      "2923141610654231562963690313126/27"
    ],
    [
      "75689751056131762924",
      "63254523617629215045094745638"
    ],
    [
      "147313922300455119077660284/1256641",
      "1056014134794145660395024383874312800918/1408694561"
    ],
    [
      "62008343726414001654",
      "113233128894864075200878584168"
    ],
    [
      "-973584631196637271691/16",
      "75214490969680849341937968549007/64"
    ],
    [
      "603564895249076687865063844/7789681",
      "-2041233295025680874121194985458394733502/21740999671"
    ],
    [
      "13222703214285837537609916/208849",
      "8921294749637897990914958183333634814/95443993"
    ]
  ],
  "submitter": "Natanael Wildner Fraga",
  "history": [],
  "commentary": "2-isogenous image of D-z6old678-077, rational kernel x=-36711517139185885084. 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 20:29:39",
  "updated_at": "2026-09-26 20:29:39"
}