{
  "checks": 428109,
  "encoding": "integer sum c_i p^i for field element sum c_i a^i; ordinary/q exponents explicitly separate",
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      "dual_basis": [
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        ],
        "dual_generator": 3,
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        "normal_elements": [
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          5,
          7
        ],
        "orbit": [
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          5,
          7
        ],
        "q": 2
      },
      "operators": {
        "exhaustive": true,
        "operators": 512,
        "rank_counts": {
          "0": 1,
          "1": 49,
          "2": 294,
          "3": 168
        }
      },
      "p": 2
    },
    {
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      "basis": [
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      "dual_basis": [
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          5,
          7,
          8
        ],
        "orbit": [
          4,
          7
        ],
        "q": 3
      },
      "operators": {
        "exhaustive": true,
        "operators": 81,
        "rank_counts": {
          "0": 1,
          "1": 32,
          "2": 48
        }
      },
      "p": 3,
      "spaces": {
        "dimensions": {
          "0": 1,
          "1": 4,
          "2": 1
        },
        "ordered_pairs": 36,
        "subspaces": 6
      }
    },
    {
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        1
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      "basis": [
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        2,
        4,
        8
      ],
      "dual_basis": [
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        1
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      "modulus": [
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        1
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        "count": 8,
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            1
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        ],
        "dual_generator": 14,
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          11,
          12,
          13,
          14,
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        ],
        "orbit": [
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          12,
          15,
          10
        ],
        "q": 2
      },
      "operators": {
        "exhaustive": false,
        "operators": 20,
        "rank_counts": {
          "2": 1,
          "3": 9,
          "4": 10
        }
      },
      "p": 2,
      "spaces": {
        "dimensions": {
          "0": 1,
          "1": 15,
          "2": 35,
          "3": 15,
          "4": 1
        },
        "ordered_pairs": 4489,
        "subspaces": 67
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    },
    {
      "base_field_labels": [
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        1,
        6,
        7
      ],
      "basis": [
        1,
        2
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      "dual_basis": [
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      "modulus": [
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        1
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        "count": 12,
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        "dual_generator": 2,
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          4,
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          11,
          12,
          13,
          14,
          15
        ],
        "orbit": [
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        "q": 4
      },
      "operators": {
        "exhaustive": true,
        "operators": 256,
        "rank_counts": {
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          "1": 75,
          "2": 180
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      },
      "p": 2,
      "spaces": {
        "dimensions": {
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          "1": 5,
          "2": 1
        },
        "ordered_pairs": 49,
        "subspaces": 7
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    },
    {
      "base_field_labels": [
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        1,
        58,
        59
      ],
      "basis": [
        1,
        2,
        4
      ],
      "dual_basis": [
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        5,
        3
      ],
      "modulus": [
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        0,
        0,
        0,
        0,
        1
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      "normal": {
        "chosen_generator": 2,
        "count": 27,
        "distinct_factors": [
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            1
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          [
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          [
            59,
            1
          ]
        ],
        "dual_generator": 12,
        "n": 3,
        "normal_elements": [
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          34,
          35,
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          42,
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          44,
          45,
          50,
          51,
          52,
          53,
          54,
          55,
          56,
          57,
          62,
          63
        ],
        "orbit": [
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          19
        ],
        "q": 4
      },
      "operators": {
        "exhaustive": false,
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        "rank_counts": {
          "2": 28,
          "3": 39
        }
      },
      "p": 2
    }
  ],
  "main_F16": {
    "P_U": [
      7,
      6,
      1
    ],
    "P_W": [
      6,
      7,
      1
    ],
    "P_W_U_image": [
      0,
      7
    ],
    "P_intersection": [
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      1
    ],
    "P_sum": [
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      1,
      1,
      1
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    "U": [
      0,
      1,
      4,
      5
    ],
    "W": [
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      3
    ],
    "a": 2,
    "a_orbit": [
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      3,
      5
    ],
    "adjoint_kernel": [
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      13
    ],
    "alpha": 8,
    "alpha_orbit": [
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      12,
      15,
      10
    ],
    "ell": [
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      1
    ],
    "ell_Dickson": [
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        2,
        0,
        1,
        0
      ],
      [
        0,
        4,
        0,
        1
      ],
      [
        1,
        0,
        3,
        0
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      [
        0,
        1,
        0,
        5
      ]
    ],
    "ell_inverse": [
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      0,
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    ],
    "ell_matrix": [
      [
        1,
        1,
        1,
        0
      ],
      [
        1,
        1,
        0,
        0
      ],
      [
        0,
        1,
        1,
        1
      ],
      [
        0,
        0,
        1,
        1
      ]
    ],
    "normal_matrix": [
      [
        0,
        0,
        1,
        0
      ],
      [
        0,
        0,
        1,
        1
      ],
      [
        0,
        1,
        1,
        0
      ],
      [
        1,
        1,
        1,
        1
      ]
    ],
    "singular": [
      2,
      1
    ],
    "singular_adjoint": [
      2,
      0,
      0,
      1
    ],
    "singular_kernel": [
      0,
      2
    ],
    "singular_matrix": [
      [
        1,
        0,
        1,
        1
      ],
      [
        1,
        0,
        1,
        1
      ],
      [
        0,
        0,
        0,
        1
      ],
      [
        0,
        0,
        1,
        1
      ]
    ]
  },
  "scope": "finite exact certificates, not general theorem proofs; full operator enumeration only for stated exhaustive cases",
  "status": "PASS",
  "unit": "algebra-A18"
}