{
  "id": 2967,
  "curve_key": "934327736298362201828744402996817123649:-27366227762719572130743973678162793136474822565809310260193",
  "ainvs": [
    "1",
    "0",
    "0",
    "-19465161172882545871432175062433690076",
    "31673874725369875149701946844577835455707993010780190480"
  ],
  "rank_lower_bound": 8,
  "torsion": [
    6
  ],
  "naive_height": 269.19867185957844,
  "faltings_height": 20.36399069644821,
  "conductor": "1315458511091655433538805332647246470",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "11",
    "19",
    "31",
    "43",
    "47",
    "59",
    "61",
    "73",
    "79",
    "101",
    "103",
    "127",
    "131",
    "149",
    "421",
    "433",
    "4903"
  ],
  "discriminant": "38615793529450755136912783555945908559130437471416139879628763515461325521751860657202183214614775809756784230400",
  "regulator": "1732690584.22923288969706676199434279845749855981721806",
  "points": [
    [
      "8334917146496073273131784/2374681",
      "-9396238970548725482036438899586865660/3659383421"
    ],
    [
      "5046724617429702011822364/982081",
      "7987015283578280216003184175158391596/973242271"
    ],
    [
      "-174317393926140171885762/96721",
      "234743899215161958475263391490123616/30080231"
    ],
    [
      "3298840421728417365166264/23409",
      "-5988685078336988949220413227531803132/3581577"
    ],
    [
      "2080122959349423480",
      "429481163334355785761928420"
    ],
    [
      "2106940935186835008",
      "122728396591884791558535516"
    ],
    [
      "582229969978843184872345954602524/9959988843025",
      "14009931316951850663474813892881835478848000280528/31433176989200533625"
    ],
    [
      "229122318433481169685199108/74114881",
      "-651775222437087268609846333069476213656/638055010529"
    ]
  ],
  "submitter": "Natanael Wildner Fraga",
  "history": [],
  "commentary": "2-isogenous image of D-z6old678-061, rational kernel x=-16875178086556225985/4. 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 20:30:47",
  "updated_at": "2026-09-26 20:30:47"
}