{
  "id": 2891,
  "curve_key": "63231086986982412896389795281:-15899957796178605067800846768273896955547929",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-1317314312228800268674787402",
    "18402728930762603712606815033717512395401"
  ],
  "rank_lower_bound": 7,
  "torsion": [
    6
  ],
  "naive_height": 198.94978072128845,
  "faltings_height": 13.84614616852942,
  "conductor": "1855529015792879526484770",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "11",
    "17",
    "19",
    "23",
    "83",
    "719",
    "56807",
    "10631519"
  ],
  "discriminant": "11349211283493786614190300840325130622696634758278789388360900169728000",
  "regulator": "23351073.44604075189124340461726080616479603215737313",
  "points": [
    [
      "545581506150889/25",
      "-866620726782491457573/125"
    ],
    [
      "20962160120209",
      "-58156797066976521"
    ],
    [
      "30296616017521",
      "79381158175387391319"
    ],
    [
      "20955105124261",
      "-2216217608086241"
    ],
    [
      "1273651704680379241/57121",
      "-146870456720892645954591039/13651919"
    ],
    [
      "203931098788240739578489/9743071849",
      "182528813095967528625642261656797/961709392999243"
    ],
    [
      "1707019240611361/81",
      "-691289034488538009569/729"
    ]
  ],
  "submitter": "Natanael Wildner Fraga",
  "history": [],
  "commentary": "2-isogenous image of D-z6old678-135, rational kernel x=-10477415138799. 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 20:27:07",
  "updated_at": "2026-09-26 20:27:07"
}