{
  "id": 2991,
  "curve_key": "381764273620716228485451575827950961:-230878974315227023267617590666301868185285457230236041",
  "ainvs": [
    "1",
    "0",
    "0",
    "-7953422367098254760113574496415645",
    "267221035087068313304364791938920747056838071167025"
  ],
  "rank_lower_bound": 8,
  "torsion": [
    6
  ],
  "naive_height": 245.79033420651717,
  "faltings_height": 18.3813349806013,
  "conductor": "4455223827985669672160352345650430",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "11",
    "13",
    "17",
    "23",
    "37",
    "79",
    "89",
    "139",
    "269",
    "409",
    "1753",
    "33119",
    "1642741"
  ],
  "discriminant": "1351120646548266910856737179168576350210738298342613665574246575612937385098494025776197285591616000000",
  "regulator": "53409781.4480988163994516708847476403167415051451924111",
  "points": [
    [
      "1292163218254234790",
      "1465433947207812873574939655"
    ],
    [
      "154413713455433126",
      "52162070278733879693923847"
    ],
    [
      "4712750430947330",
      "15160580813180931931243055"
    ],
    [
      "152566862651188067396/3721",
      "719466494376242882201469607217/226981"
    ],
    [
      "41033019185609029216851410/61105489",
      "-260633950869302010123432314511867640985/477661607513"
    ],
    [
      "312954106606284232490/5041",
      "-1277027468301964716062429204495/357911"
    ],
    [
      "67760242934248922",
      "-6277977905739570075526009"
    ],
    [
      "1022063445780343639730/17161",
      "-4920598597114467422556998361995/2248091"
    ]
  ],
  "submitter": "Natanael Wildner Fraga",
  "history": [],
  "commentary": "2-isogenous image of D-z6old678-111, rational kernel x=-361915575570613521/4. 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 20:32:15",
  "updated_at": "2026-09-26 20:32:15"
}