{
  "id": 3026,
  "curve_key": "8039584219562551187212615141249:-20848028049457839894709770333595151453357288257",
  "ainvs": [
    "1",
    "0",
    "0",
    "-167491337907553149733596148776",
    "24129662094279893327932223367787466012077056"
  ],
  "rank_lower_bound": 8,
  "torsion": [
    6
  ],
  "naive_height": 213.48579047307084,
  "faltings_height": 15.756424775392672,
  "conductor": "29700487374913942444634217870",
  "bad_primes": [
    "2",
    "3",
    "5",
    "11",
    "13",
    "17",
    "19",
    "23",
    "53",
    "83",
    "139",
    "163",
    "283",
    "401",
    "859",
    "95917"
  ],
  "discriminant": "49188405370646308675681916615228543581270463621534619250434652753148207090437046640974400",
  "regulator": "2590324.16777865123719040906312665383245137207124019552",
  "points": [
    [
      "87242841500652010/81",
      "24103609513125664253483474/729"
    ],
    [
      "154233749714574",
      "1402057369900464232548"
    ],
    [
      "70835303689912186806/235225",
      "-114091836077787524518737156834/114084125"
    ],
    [
      "-276816607335726",
      "7020135441693797788308"
    ],
    [
      "-2740727554175139/16",
      "442452489937165905446163/64"
    ],
    [
      "4359465102120925496908854/96963409",
      "-9101896095553260317349749627331827094/954798688423"
    ],
    [
      "1874871015967568116234454682/14825480049769",
      "127269345569901656457037877692854876239286/57083835652389910603"
    ],
    [
      "34274726570413695559426554/3073482721",
      "-200526664807247253590388018108563835348/170390808569519"
    ]
  ],
  "submitter": "Natanael Wildner Fraga",
  "history": [],
  "commentary": "2-isogenous image of D-z6old678-050, rational kernel x=-88697342525360. 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 20:33:51",
  "updated_at": "2026-09-26 20:33:51"
}