{
  "id": 3052,
  "curve_key": "410941917746428372884230368:-8330473631798626079016957262059107075296",
  "ainvs": [
    "0",
    "1",
    "0",
    "-8561289953050591101754799",
    "9641751888652963457766757745164122330"
  ],
  "rank_lower_bound": 8,
  "torsion": [
    4
  ],
  "naive_height": 183.84148235104516,
  "faltings_height": 12.909037303362398,
  "conductor": "1605314370710886136520856",
  "bad_primes": [
    "2",
    "3",
    "7",
    "13",
    "29",
    "31",
    "47",
    "151",
    "181",
    "313",
    "661",
    "1009",
    "3049"
  ],
  "discriminant": "179567262762147047862570354209867984544986573296506572400834672648453072",
  "regulator": "12754728.1889733735203215859888936732113911641919580549",
  "points": [
    [
      "679112366659555/289",
      "-7767440657668097623305/4913"
    ],
    [
      "1692165031114",
      "-4451969470616028"
    ],
    [
      "205485798616954/121",
      "-26031189376420418988/1331"
    ],
    [
      "2609718940795",
      "-2252338595508577905"
    ],
    [
      "1068320186561566/169",
      "-31695373582268857499934/2197"
    ],
    [
      "76941927682620246649/42042256",
      "87590050267843650911670375957/272601987904"
    ],
    [
      "154220487318292447/80089",
      "-12334688929867619124981441/22665187"
    ],
    [
      "1671818001295",
      "39031615111212045"
    ]
  ],
  "submitter": "Natanael Wildner Fraga",
  "history": [],
  "commentary": "2-isogenous image of D-z2z4old678-023, rational kernel x=-3382740935541. Source: https://dujella.github.io/tors/z2z4old678.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 20:35:02",
  "updated_at": "2026-09-26 20:35:02"
}