{
  "exact_presentation": {
    "order": "O_sing = Z_3 e7 \u2295 Z_3 epair \u2295 Z_3 n",
    "relations": [
      "e7^2 = e7",
      "epair^2 = epair",
      "e7*epair = 0",
      "e7*n = 0",
      "epair*n = n",
      "n^2 = 3n"
    ]
  },
  "jacobson_radical": {
    "J": [
      "3 e7",
      "3 epair",
      "n"
    ],
    "J_formula_for_r_ge_1": "J^r = Z_3\u00b73^r e7 \u2295 Z_3\u00b73^r epair \u2295 Z_3\u00b73^(r-1) n",
    "O_mod_J": [
      "e7 mod J",
      "epair mod J"
    ],
    "O_mod_J_dimension_F3": 2,
    "J_mod_3O": [
      "n mod 3O"
    ],
    "J_mod_3O_dimension_F3": 1,
    "O_mod_3O_dimension_F3": 3,
    "layers": [
      {
        "r": 1,
        "Jr_generators": [
          "3^1 e7",
          "3^1 epair",
          "n"
        ],
        "Jr_plus_1_generators": [
          "3^2 e7",
          "3^2 epair",
          "3^1 n"
        ],
        "quotient_basis": [
          "3^1 e7",
          "3^1 epair",
          "n"
        ],
        "quotient_dimension_F3": 3
      },
      {
        "r": 2,
        "Jr_generators": [
          "3^2 e7",
          "3^2 epair",
          "3^1 n"
        ],
        "Jr_plus_1_generators": [
          "3^3 e7",
          "3^3 epair",
          "3^2 n"
        ],
        "quotient_basis": [
          "3^2 e7",
          "3^2 epair",
          "3^1 n"
        ],
        "quotient_dimension_F3": 3
      },
      {
        "r": 3,
        "Jr_generators": [
          "3^3 e7",
          "3^3 epair",
          "3^2 n"
        ],
        "Jr_plus_1_generators": [
          "3^4 e7",
          "3^4 epair",
          "3^3 n"
        ],
        "quotient_basis": [
          "3^3 e7",
          "3^3 epair",
          "3^2 n"
        ],
        "quotient_dimension_F3": 3
      },
      {
        "r": 4,
        "Jr_generators": [
          "3^4 e7",
          "3^4 epair",
          "3^3 n"
        ],
        "Jr_plus_1_generators": [
          "3^5 e7",
          "3^5 epair",
          "3^4 n"
        ],
        "quotient_basis": [
          "3^4 e7",
          "3^4 epair",
          "3^3 n"
        ],
        "quotient_dimension_F3": 3
      },
      {
        "r": 5,
        "Jr_generators": [
          "3^5 e7",
          "3^5 epair",
          "3^4 n"
        ],
        "Jr_plus_1_generators": [
          "3^6 e7",
          "3^6 epair",
          "3^5 n"
        ],
        "quotient_basis": [
          "3^5 e7",
          "3^5 epair",
          "3^4 n"
        ],
        "quotient_dimension_F3": 3
      },
      {
        "r": 6,
        "Jr_generators": [
          "3^6 e7",
          "3^6 epair",
          "3^5 n"
        ],
        "Jr_plus_1_generators": [
          "3^7 e7",
          "3^7 epair",
          "3^6 n"
        ],
        "quotient_basis": [
          "3^6 e7",
          "3^6 epair",
          "3^5 n"
        ],
        "quotient_dimension_F3": 3
      }
    ]
  },
  "associated_graded_multiplication": {
    "notation": {
      "alpha_r": "class of 3^r e7 in J^r/J^(r+1)",
      "beta_r": "class of 3^r epair in J^r/J^(r+1)",
      "gamma_r": "class of 3^(r-1) n in J^r/J^(r+1)"
    },
    "products": [
      "alpha_r * alpha_s = alpha_(r+s)",
      "beta_r * beta_s = beta_(r+s)",
      "alpha_r * beta_s = 0",
      "alpha_r * gamma_s = 0",
      "beta_r * gamma_s = gamma_(r+s)",
      "gamma_r * gamma_s = gamma_(r+s)"
    ]
  }
}