{
  "id": 2989,
  "curve_key": "13950484002790813026359058137283211489:-94250862046726685738775662707602307273722926283901090513",
  "ainvs": [
    "1",
    "0",
    "0",
    "-290635083391475271382480377860066906",
    "109086645887415145506770760073398249692342985043954020"
  ],
  "rank_lower_bound": 8,
  "torsion": [
    6
  ],
  "naive_height": 257.77110998869824,
  "faltings_height": 19.522206698712296,
  "conductor": "3808846288107559247843042095265490",
  "bad_primes": [
    "2",
    "3",
    "5",
    "11",
    "13",
    "17",
    "23",
    "41",
    "47",
    "61",
    "73",
    "83",
    "229",
    "5297",
    "8311",
    "12589",
    "25121"
  ],
  "discriminant": "-3569581913939795835997155844150593780538971022332362278017744221776643606121783131922966479177111138535360000",
  "regulator": "9158024916.44782972458412998455555012122390490942159593",
  "points": [
    [
      "-3767907115522726193204/16129",
      "-830121082603601301352572112092350/2048383"
    ],
    [
      "449819005653722697466927728/1211805721",
      "9651790493163744918806665403538798096846/42184168953731"
    ],
    [
      "-19706032594169077904/49",
      "-137596140756980055663109728230/343"
    ],
    [
      "25332499612387068216964/63001",
      "3783134988568558970592666839756066/15813251"
    ],
    [
      "-661673219306550805397076/1901641",
      "1075127869341904164209137110119175474/2622362939"
    ],
    [
      "-14449026451492185845616613499936/61497544309369",
      "-195542146831001422715289339739673609568088161950/482266017883211144653"
    ],
    [
      "-32110667471192912733372/212521",
      "-37887566255166288203004026916913554/97972181"
    ],
    [
      "2700985152934516456547458084/11773335025",
      "298186146754945266551759530389181777218838/1277465716887625"
    ]
  ],
  "submitter": "Natanael Wildner Fraga",
  "history": [],
  "commentary": "2-isogenous image of D-z6old678-023, rational kernel x=336435780064188244. 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 20:32:11",
  "updated_at": "2026-09-26 20:32:11"
}