{
  "id": 3053,
  "curve_key": "-248307143717264106151907312:-31587906710581437645840321190531425388096",
  "ainvs": [
    "0",
    "1",
    "0",
    "5173065494109668878164736",
    "36560077211322832741553963860148538000"
  ],
  "rank_lower_bound": 8,
  "torsion": [
    4
  ],
  "naive_height": 186.5071859501609,
  "faltings_height": 13.602184483922343,
  "conductor": "1605314370710886136520856",
  "bad_primes": [
    "2",
    "3",
    "7",
    "13",
    "29",
    "31",
    "47",
    "151",
    "181",
    "313",
    "661",
    "1009",
    "3049"
  ],
  "discriminant": "-586287953862184297173491668600362427662142067465377219320191579801443220149248",
  "regulator": "204075651.023573976325145375822298771382258627071328879",
  "points": [
    [
      "8005979673136/25",
      "773077626198404216316/125"
    ],
    [
      "447176857804821/4",
      "9458331276780524434839/8"
    ],
    [
      "109695168623004979/6241",
      "36755342162372551265315922/493039"
    ],
    [
      "-473838675552233600/4853209",
      "64197881107441293152439790620/10691619427"
    ],
    [
      "7235665272505600/6241",
      "3274758322973840973562740/493039"
    ],
    [
      "13250637035742211/5329",
      "3131441715915107961542526/389017"
    ],
    [
      "21848641838219959/841",
      "3245215452195114759382812/24389"
    ],
    [
      "1412364778696725525/2116",
      "-1678505876527649130065425545/97336"
    ]
  ],
  "submitter": "Natanael Wildner Fraga",
  "history": [],
  "commentary": "2-isogenous image of D-z2z4old678-023, rational kernel x=1402149274312. 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 20:35:06",
  "updated_at": "2026-09-26 20:35:06"
}