{
  "id": 2988,
  "curve_key": "696979190462973682061608683765691561:-578351322338067371497061170256615051853307152521633941",
  "ainvs": [
    "1",
    "0",
    "1",
    "-14520399801311951709616847578451908",
    "669388104557948345430083778428419593473257091288306"
  ],
  "rank_lower_bound": 8,
  "torsion": [
    6
  ],
  "naive_height": 247.5961908697638,
  "faltings_height": 18.47681233640863,
  "conductor": "10893096871712594532803840282233410",
  "bad_primes": [
    "2",
    "3",
    "5",
    "11",
    "13",
    "17",
    "19",
    "53",
    "79",
    "103",
    "109",
    "139",
    "541",
    "881",
    "395449",
    "6383317"
  ],
  "discriminant": "2365910575324213424813059887140281073007254916373578823326758516049737704712532211172619024030842250000",
  "regulator": "871286209.097747641201563153731438860528025668943951093",
  "points": [
    [
      "1092183078002612626",
      "1134740583352845938272913594"
    ],
    [
      "97398751043592226",
      "13382607248208989778944474"
    ],
    [
      "159612118109445565",
      "49173561577405003708766267"
    ],
    [
      "-86371900913352463394210/1203409",
      "48358073772299122493929884053823106/1320139673"
    ],
    [
      "212505406183281638330965/17131321",
      "1571469583620263628472679862078269473/70906537619"
    ],
    [
      "-548637234599776003966010/25512601",
      "4016962754908996964789274463405241542/128864147651"
    ],
    [
      "91104670638338325961165/48841",
      "-27442606796538030984702731472105713/10793861"
    ],
    [
      "-30739813889800139880833710064540/783918074140369",
      "753472249017791376142614887102965025779801276726/21948559203787864640153"
    ]
  ],
  "submitter": "Natanael Wildner Fraga",
  "history": [],
  "commentary": "2-isogenous image of D-z6old678-078, rational kernel x=-591567337254347721/4. 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 20:32:04",
  "updated_at": "2026-09-26 20:32:04"
}