{
  "id": 2970,
  "curve_key": "772994076662426432365894802918425525665:-21489758195945269338279126217032834838962321238041071817201",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-16104043263800550674289475060800531785",
    "24872405319381098775182406900478881695149129457377299401"
  ],
  "rank_lower_bound": 8,
  "torsion": [
    6
  ],
  "naive_height": 268.6300042006576,
  "faltings_height": 20.072687513658252,
  "conductor": "144739364313004453827525038927796126",
  "bad_primes": [
    "2",
    "3",
    "7",
    "13",
    "17",
    "19",
    "29",
    "31",
    "41",
    "89",
    "109",
    "139",
    "173",
    "313",
    "2281",
    "2339",
    "2903",
    "6563"
  ],
  "discriminant": "40272953261685938927578784702341463563085268547081462240476624679162113271561435533081989288994877158413303808",
  "regulator": "102478392.624714793997852380575499306034321547221208859",
  "points": [
    [
      "136350592450299414416375671/51624225",
      "324065411717538709957993607248065835564/370920056625"
    ],
    [
      "5167736285524917793558591/1885129",
      "2981156009749187681042798335550933996/2588282117"
    ],
    [
      "66558368728789616817691/18225",
      "9455147933400573233187659996308456/2460375"
    ],
    [
      "2140625267896547607",
      "456749749203100883185061500"
    ],
    [
      "811665033145819500001543/199809",
      "-459662201547522492075947638446330940/89314623"
    ],
    [
      "2156229249005719109924583/954529",
      "136023912863404745191786643031598460/932574833"
    ],
    [
      "608301332889029030157136296612369999/445217216229241",
      "-474434520081036295782641340252997690475542913662783628/9394159394580960300211"
    ],
    [
      "62526770471880377755171282348047/11655798855481",
      "-383459257501677936301500052001072898122069522572/39793584984744607379"
    ]
  ],
  "submitter": "Natanael Wildner Fraga",
  "history": [],
  "commentary": "2-isogenous image of D-z6old678-095, rational kernel x=-18666384651289037877/4. 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 20:31:01",
  "updated_at": "2026-09-26 20:31:01"
}