{
  "id": 2898,
  "curve_key": "13124572642775380418294411281:1089387481413286493581159938442471196230279",
  "ainvs": [
    "1",
    "1",
    "1",
    "-273428596724487092047800235",
    "-1260865140524751073930533279856028245735"
  ],
  "rank_lower_bound": 7,
  "torsion": [
    2,
    2
  ],
  "naive_height": 194.2328512747091,
  "faltings_height": 14.210397082644525,
  "conductor": "23959330209511704615223770",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "11",
    "17",
    "19",
    "23",
    "31",
    "41",
    "43",
    "97",
    "193",
    "241",
    "2087",
    "2713"
  ],
  "discriminant": "621528005658546166397145444428078137593428577802111230401472503454441061776614400",
  "regulator": "3537273.104264385143936780341026460054517525105461388",
  "points": [
    [
      "-5450206960797",
      "-8214632578091546042"
    ],
    [
      "-8042370591997",
      "-20444326455595091562"
    ],
    [
      "-19467497201816965/2209",
      "-2237275045696324381194438/103823"
    ],
    [
      "-302099846091853/49",
      "-4734839221643410091766/343"
    ],
    [
      "39522109920803",
      "-222858907300697422842"
    ],
    [
      "366159024260575/9",
      "-6329540314689979670582/27"
    ],
    [
      "1045312272628687/9",
      "-33438288140499352950194/27"
    ]
  ],
  "submitter": "Natanael Wildner Fraga",
  "history": [],
  "commentary": "2-isogenous image of D-z2z4old678-054, rational kernel x=6694286189203. Source: https://dujella.github.io/tors/z2z4old678.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 20:27:25",
  "updated_at": "2026-09-26 20:27:25"
}