{
  "id": 2952,
  "curve_key": "105207015187015469702518189568900287839265:-34935346263963438501806057936430889125381029743074464384979889",
  "ainvs": [
    "1",
    "0",
    "0",
    "-2191812816396155618802462282685422663318",
    "40434428546253979747278064136169405000957468123519807816228"
  ],
  "rank_lower_bound": 8,
  "torsion": [
    6
  ],
  "naive_height": 283.417209366523,
  "faltings_height": 21.52073187488739,
  "conductor": "31206133418171078552200701249480523698",
  "bad_primes": [
    "2",
    "3",
    "7",
    "13",
    "17",
    "19",
    "31",
    "37",
    "41",
    "43",
    "59",
    "89",
    "97",
    "103",
    "241",
    "2179",
    "4363",
    "9349",
    "77867"
  ],
  "discriminant": "-32403288764572216931414588263712638049740601000366815099391473313771046682830609674059662848714421123093613254722360832",
  "regulator": "358821411.898662606422013888483693292567617787643322601",
  "points": [
    [
      "197547378825372012240543252/5669161",
      "1077328151117331710958814845957071546178/13498272341"
    ],
    [
      "12187465315262662023217713294/45495025",
      "1326187970352916428834669361813308455781582/306863943625"
    ],
    [
      "219173891807937428377075961262/7745584081",
      "22304646157851209166676353575010107950355892/681681109384729"
    ],
    [
      "848516061670468129121922085302/6196680961",
      "741027408334122281276776159366530147363663540/487796528568959"
    ],
    [
      "32616167912949789870541294086/1659666121",
      "-4757146412030640116956789443092627752189302/67613138103419"
    ],
    [
      "-29685007250922343695825536064/828000625",
      "-6434413391983354586445226176394330550728266/23825717984375"
    ],
    [
      "19829617797827890931288132951476479673639394664143078861480214/37839954002796389975258106067756400087041",
      "-87961561305317918747478189995530840375920498408288545463619444319515239599358447165836681580/7360814949365290411721864974032032103217419155525365598627839"
    ],
    [
      "-133997709194063451913582276120558387290583120056945264738/2578561569062255695772961349780080481",
      "-489950108797155448661918604335260782437306644184991177395655515639621134021016314692/4140628418084450332678003652314874689518425305375573871"
    ]
  ],
  "submitter": "Natanael Wildner Fraga",
  "history": [],
  "commentary": "2-isogenous image of D-z6old678-117, rational kernel x=27100800076051676620. 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 20:29:50",
  "updated_at": "2026-09-26 20:29:50"
}