{
  "id": 3004,
  "curve_key": "120771645722157912539052916369987510441:-1327204216056835150303591920273237350609852293665511331989",
  "ainvs": [
    "1",
    "0",
    "1",
    "-2516075952544956511230269091041406468",
    "1536115990806522164493558393159722779862584397118669058"
  ],
  "rank_lower_bound": 8,
  "torsion": [
    6
  ],
  "naive_height": 263.06089465467613,
  "faltings_height": 19.572602475186514,
  "conductor": "24971493095101000569198256701990",
  "bad_primes": [
    "2",
    "3",
    "5",
    "13",
    "17",
    "23",
    "29",
    "41",
    "53",
    "97",
    "103",
    "211",
    "281",
    "971",
    "22697",
    "199049"
  ],
  "discriminant": "45648673656937852714764289378032721541822788395045649967436068606968512448237175626474375396826640036000000",
  "regulator": "50803823.7302572451427685133835350015062051096179421517",
  "points": [
    [
      "179621414078996820740269049/145467721",
      "979960538951022401902535759553502955115/1754486182981"
    ],
    [
      "264982684761137637216721/91809",
      "119058874981790130394055219712687505/27818127"
    ],
    [
      "4517791691032554558019442/4897369",
      "102218407677512192174475330841895586/10837877597"
    ],
    [
      "-377715091563866441211439/1181569",
      "1951123570508293699510497788446119345/1284365503"
    ],
    [
      "481065332898252689",
      "661096867632681894737885775"
    ],
    [
      "63919824419817873761/49",
      "-236086000977109153286878839495/343"
    ],
    [
      "2131803731923622369",
      "-2420847731689879709777596785"
    ],
    [
      "151278835491836821775340039815044241/163541139364840000",
      "-935029617107943703559170227730062893107108636574061/66136396549647898648000000"
    ]
  ],
  "submitter": "Natanael Wildner Fraga",
  "history": [],
  "commentary": "2-isogenous image of D-z6old678-116, rational kernel x=-7298086438714790153/4. 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 20:32:54",
  "updated_at": "2026-09-26 20:32:54"
}