{
  "id": 2972,
  "curve_key": "24868351648983011748425169094173555169:-304252792378871450299719658642925680176291795218909212753",
  "ainvs": [
    "1",
    "0",
    "0",
    "-518090659353812744758857689461949066",
    "352144435623693808174093901816494401103766251825982500"
  ],
  "rank_lower_bound": 8,
  "torsion": [
    6
  ],
  "naive_height": 260.1149078628263,
  "faltings_height": 19.72686263899069,
  "conductor": "2059735484183469504222249000949231390",
  "bad_primes": [
    "2",
    "5",
    "7",
    "13",
    "19",
    "31",
    "37",
    "41",
    "47",
    "59",
    "61",
    "67",
    "101",
    "139",
    "461",
    "1231",
    "1867",
    "15027101"
  ],
  "discriminant": "-44670315256660085289614946510325585562907075236420454898967729067161240142828390265413196085845580078016102400",
  "regulator": "71205929.5240556216203660310872410911243294583281651870",
  "points": [
    [
      "220882760547855678904/441",
      "4326993703625014680685542454970/9261"
    ],
    [
      "656892175423109748",
      "-543386165165085342991160474"
    ],
    [
      "36462052065818001080676/100489",
      "14664678745856608094794122214195438/31855013"
    ],
    [
      "-195452771575426412",
      "-667787425648837458940452954"
    ],
    [
      "4753201458650490400284052/16297369",
      "-31266990169747230761963535682283883762/65792478653"
    ],
    [
      "8974988097547702847441428/241081",
      "-26882590415070742185921196331936848814/118370771"
    ],
    [
      "111766661434996149069347056/36517849",
      "-1155895512438792344808199162085040688098/220677361507"
    ],
    [
      "167728567372966492852/1089",
      "-18879805775339694980913402292378/35937"
    ]
  ],
  "submitter": "Natanael Wildner Fraga",
  "history": [],
  "commentary": "2-isogenous image of D-z6old678-056, rational kernel x=7551961523951506015/4. 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 20:31:05",
  "updated_at": "2026-09-26 20:31:05"
}