{
  "id": 2959,
  "curve_key": "723999090815658924804132467450657846586561:-960033326762298741996694053008065558646113769170088732510001441",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-15083314391992894266752759738555371803887",
    "1111149683752660581018463019579555583110801348696315616689399"
  ],
  "rank_lower_bound": 8,
  "torsion": [
    6
  ],
  "naive_height": 290.0441471577589,
  "faltings_height": 22.202848943102566,
  "conductor": "2487091744114190027859845240035171290",
  "bad_primes": [
    "2",
    "3",
    "5",
    "11",
    "13",
    "17",
    "23",
    "41",
    "59",
    "73",
    "83",
    "97",
    "103",
    "109",
    "127",
    "151",
    "1327",
    "7823",
    "155539"
  ],
  "discriminant": "-313751154056900904331587271195142929622844082602615161034254118972326404596543980472870071663616277129979756544000000000000",
  "regulator": "228939734.934822854988518085377795657111981502541567107",
  "points": [
    [
      "865841141442688857163/9",
      "20032288537164037358396295203842/27"
    ],
    [
      "11078899071388059922243/49",
      "1043718918559427129528228575705578/343"
    ],
    [
      "110201848120231385005197509448523/838171377289",
      "-908227847977799168842665082304338827855666422402/767360144821493413"
    ],
    [
      "6487057577624565465212480563/199809",
      "522478626021906410934003600496379442151258/89314623"
    ],
    [
      "-60212137725683728549953/3721",
      "-263824414981936285581175383152310674/226981"
    ],
    [
      "22618129532002434797386192203/138603529",
      "-2824129351876003621827199558035363786076418/1631779346917"
    ],
    [
      "-5777706787156060429973/361",
      "-7964874449543668830378924464638686/6859"
    ],
    [
      "-206556124286467882981211677413/21910216441",
      "-3629612571757029358155790590052981145302731794/3243172147813261"
    ]
  ],
  "submitter": "Natanael Wildner Fraga",
  "history": [],
  "commentary": "2-isogenous image of D-z6old678-086, rational kernel x=1199793431050579939947/4. 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 20:30:16",
  "updated_at": "2026-09-26 20:30:16"
}