{
  "pair_factor": {
    "ring": "R = {(a,b) in Z_3 \u00d7 Z_3 : a \u2261 b mod 3}",
    "maximal_ideal": "m = 3(Z_3 \u00d7 Z_3)",
    "radical_power_formula": "m^r = 3^r(Z_3 \u00d7 Z_3)",
    "finite_quotient": "R_k = R / 3^k R"
  },
  "classification": {
    "statement": [
      "For k >= 1, every ideal of R_k is either a radical power m^r (0 <= r <= k),",
      "or, for some 1 <= r < k and some lambda in P^1(F_3),",
      "the line lift I_{r,lambda} = m^(r+1) + (Z/3^k Z)\u00b7v_{r,lambda},",
      "where v_{r,lambda} is any lift of the corresponding line in m^r/m^(r+1) \u2245 F_3^2."
    ],
    "line_generators": {
      "0": "(3^r,0)",
      "1": "(3^r,3^r) = 3^r\u00b7(1,1)",
      "2": "(3^r,-3^r)",
      "inf": "(0,3^r)"
    }
  },
  "finite_levels": {
    "3": {
      "k": 1,
      "chain_ideals": [
        "m^0",
        "m^1"
      ],
      "line_ideals": [],
      "ideal_count_formula": 2
    },
    "9": {
      "k": 2,
      "chain_ideals": [
        "m^0",
        "m^1",
        "m^2"
      ],
      "line_ideals": [
        {
          "name": "I_{1,0}",
          "depth": 1,
          "lambda": "0",
          "formula": "m^2 + Z_3\u00b7(3^1, 0*3^1)",
          "size": 3
        },
        {
          "name": "I_{1,1}",
          "depth": 1,
          "lambda": "1",
          "formula": "m^2 + Z_3\u00b7(3^1, 1*3^1)",
          "size": 3
        },
        {
          "name": "I_{1,2}",
          "depth": 1,
          "lambda": "2",
          "formula": "m^2 + Z_3\u00b7(3^1, 2*3^1)",
          "size": 3
        },
        {
          "name": "I_{1,\u221e}",
          "depth": 1,
          "lambda": "inf",
          "formula": "m^2 + Z_3\u00b7(0, 3^1)",
          "size": 3
        }
      ],
      "ideal_count_formula": 7
    },
    "27": {
      "k": 3,
      "chain_ideals": [
        "m^0",
        "m^1",
        "m^2",
        "m^3"
      ],
      "line_ideals": [
        {
          "name": "I_{1,0}",
          "depth": 1,
          "lambda": "0",
          "formula": "m^2 + Z_3\u00b7(3^1, 0*3^1)",
          "size": 27
        },
        {
          "name": "I_{1,1}",
          "depth": 1,
          "lambda": "1",
          "formula": "m^2 + Z_3\u00b7(3^1, 1*3^1)",
          "size": 27
        },
        {
          "name": "I_{1,2}",
          "depth": 1,
          "lambda": "2",
          "formula": "m^2 + Z_3\u00b7(3^1, 2*3^1)",
          "size": 27
        },
        {
          "name": "I_{1,\u221e}",
          "depth": 1,
          "lambda": "inf",
          "formula": "m^2 + Z_3\u00b7(0, 3^1)",
          "size": 27
        },
        {
          "name": "I_{2,0}",
          "depth": 2,
          "lambda": "0",
          "formula": "m^3 + Z_3\u00b7(3^2, 0*3^2)",
          "size": 3
        },
        {
          "name": "I_{2,1}",
          "depth": 2,
          "lambda": "1",
          "formula": "m^3 + Z_3\u00b7(3^2, 1*3^2)",
          "size": 3
        },
        {
          "name": "I_{2,2}",
          "depth": 2,
          "lambda": "2",
          "formula": "m^3 + Z_3\u00b7(3^2, 2*3^2)",
          "size": 3
        },
        {
          "name": "I_{2,\u221e}",
          "depth": 2,
          "lambda": "inf",
          "formula": "m^3 + Z_3\u00b7(0, 3^2)",
          "size": 3
        }
      ],
      "ideal_count_formula": 12
    },
    "81": {
      "k": 4,
      "chain_ideals": [
        "m^0",
        "m^1",
        "m^2",
        "m^3",
        "m^4"
      ],
      "line_ideals": [
        {
          "name": "I_{1,0}",
          "depth": 1,
          "lambda": "0",
          "formula": "m^2 + Z_3\u00b7(3^1, 0*3^1)",
          "size": 243
        },
        {
          "name": "I_{1,1}",
          "depth": 1,
          "lambda": "1",
          "formula": "m^2 + Z_3\u00b7(3^1, 1*3^1)",
          "size": 243
        },
        {
          "name": "I_{1,2}",
          "depth": 1,
          "lambda": "2",
          "formula": "m^2 + Z_3\u00b7(3^1, 2*3^1)",
          "size": 243
        },
        {
          "name": "I_{1,\u221e}",
          "depth": 1,
          "lambda": "inf",
          "formula": "m^2 + Z_3\u00b7(0, 3^1)",
          "size": 243
        },
        {
          "name": "I_{2,0}",
          "depth": 2,
          "lambda": "0",
          "formula": "m^3 + Z_3\u00b7(3^2, 0*3^2)",
          "size": 27
        },
        {
          "name": "I_{2,1}",
          "depth": 2,
          "lambda": "1",
          "formula": "m^3 + Z_3\u00b7(3^2, 1*3^2)",
          "size": 27
        },
        {
          "name": "I_{2,2}",
          "depth": 2,
          "lambda": "2",
          "formula": "m^3 + Z_3\u00b7(3^2, 2*3^2)",
          "size": 27
        },
        {
          "name": "I_{2,\u221e}",
          "depth": 2,
          "lambda": "inf",
          "formula": "m^3 + Z_3\u00b7(0, 3^2)",
          "size": 27
        },
        {
          "name": "I_{3,0}",
          "depth": 3,
          "lambda": "0",
          "formula": "m^4 + Z_3\u00b7(3^3, 0*3^3)",
          "size": 3
        },
        {
          "name": "I_{3,1}",
          "depth": 3,
          "lambda": "1",
          "formula": "m^4 + Z_3\u00b7(3^3, 1*3^3)",
          "size": 3
        },
        {
          "name": "I_{3,2}",
          "depth": 3,
          "lambda": "2",
          "formula": "m^4 + Z_3\u00b7(3^3, 2*3^3)",
          "size": 3
        },
        {
          "name": "I_{3,\u221e}",
          "depth": 3,
          "lambda": "inf",
          "formula": "m^4 + Z_3\u00b7(0, 3^3)",
          "size": 3
        }
      ],
      "ideal_count_formula": 17
    },
    "243": {
      "k": 5,
      "chain_ideals": [
        "m^0",
        "m^1",
        "m^2",
        "m^3",
        "m^4",
        "m^5"
      ],
      "line_ideals": [
        {
          "name": "I_{1,0}",
          "depth": 1,
          "lambda": "0",
          "formula": "m^2 + Z_3\u00b7(3^1, 0*3^1)",
          "size": 2187
        },
        {
          "name": "I_{1,1}",
          "depth": 1,
          "lambda": "1",
          "formula": "m^2 + Z_3\u00b7(3^1, 1*3^1)",
          "size": 2187
        },
        {
          "name": "I_{1,2}",
          "depth": 1,
          "lambda": "2",
          "formula": "m^2 + Z_3\u00b7(3^1, 2*3^1)",
          "size": 2187
        },
        {
          "name": "I_{1,\u221e}",
          "depth": 1,
          "lambda": "inf",
          "formula": "m^2 + Z_3\u00b7(0, 3^1)",
          "size": 2187
        },
        {
          "name": "I_{2,0}",
          "depth": 2,
          "lambda": "0",
          "formula": "m^3 + Z_3\u00b7(3^2, 0*3^2)",
          "size": 243
        },
        {
          "name": "I_{2,1}",
          "depth": 2,
          "lambda": "1",
          "formula": "m^3 + Z_3\u00b7(3^2, 1*3^2)",
          "size": 243
        },
        {
          "name": "I_{2,2}",
          "depth": 2,
          "lambda": "2",
          "formula": "m^3 + Z_3\u00b7(3^2, 2*3^2)",
          "size": 243
        },
        {
          "name": "I_{2,\u221e}",
          "depth": 2,
          "lambda": "inf",
          "formula": "m^3 + Z_3\u00b7(0, 3^2)",
          "size": 243
        },
        {
          "name": "I_{3,0}",
          "depth": 3,
          "lambda": "0",
          "formula": "m^4 + Z_3\u00b7(3^3, 0*3^3)",
          "size": 27
        },
        {
          "name": "I_{3,1}",
          "depth": 3,
          "lambda": "1",
          "formula": "m^4 + Z_3\u00b7(3^3, 1*3^3)",
          "size": 27
        },
        {
          "name": "I_{3,2}",
          "depth": 3,
          "lambda": "2",
          "formula": "m^4 + Z_3\u00b7(3^3, 2*3^3)",
          "size": 27
        },
        {
          "name": "I_{3,\u221e}",
          "depth": 3,
          "lambda": "inf",
          "formula": "m^4 + Z_3\u00b7(0, 3^3)",
          "size": 27
        },
        {
          "name": "I_{4,0}",
          "depth": 4,
          "lambda": "0",
          "formula": "m^5 + Z_3\u00b7(3^4, 0*3^4)",
          "size": 3
        },
        {
          "name": "I_{4,1}",
          "depth": 4,
          "lambda": "1",
          "formula": "m^5 + Z_3\u00b7(3^4, 1*3^4)",
          "size": 3
        },
        {
          "name": "I_{4,2}",
          "depth": 4,
          "lambda": "2",
          "formula": "m^5 + Z_3\u00b7(3^4, 2*3^4)",
          "size": 3
        },
        {
          "name": "I_{4,\u221e}",
          "depth": 4,
          "lambda": "inf",
          "formula": "m^5 + Z_3\u00b7(0, 3^4)",
          "size": 3
        }
      ],
      "ideal_count_formula": 22
    },
    "729": {
      "k": 6,
      "chain_ideals": [
        "m^0",
        "m^1",
        "m^2",
        "m^3",
        "m^4",
        "m^5",
        "m^6"
      ],
      "line_ideals": [
        {
          "name": "I_{1,0}",
          "depth": 1,
          "lambda": "0",
          "formula": "m^2 + Z_3\u00b7(3^1, 0*3^1)",
          "size": 19683
        },
        {
          "name": "I_{1,1}",
          "depth": 1,
          "lambda": "1",
          "formula": "m^2 + Z_3\u00b7(3^1, 1*3^1)",
          "size": 19683
        },
        {
          "name": "I_{1,2}",
          "depth": 1,
          "lambda": "2",
          "formula": "m^2 + Z_3\u00b7(3^1, 2*3^1)",
          "size": 19683
        },
        {
          "name": "I_{1,\u221e}",
          "depth": 1,
          "lambda": "inf",
          "formula": "m^2 + Z_3\u00b7(0, 3^1)",
          "size": 19683
        },
        {
          "name": "I_{2,0}",
          "depth": 2,
          "lambda": "0",
          "formula": "m^3 + Z_3\u00b7(3^2, 0*3^2)",
          "size": 2187
        },
        {
          "name": "I_{2,1}",
          "depth": 2,
          "lambda": "1",
          "formula": "m^3 + Z_3\u00b7(3^2, 1*3^2)",
          "size": 2187
        },
        {
          "name": "I_{2,2}",
          "depth": 2,
          "lambda": "2",
          "formula": "m^3 + Z_3\u00b7(3^2, 2*3^2)",
          "size": 2187
        },
        {
          "name": "I_{2,\u221e}",
          "depth": 2,
          "lambda": "inf",
          "formula": "m^3 + Z_3\u00b7(0, 3^2)",
          "size": 2187
        },
        {
          "name": "I_{3,0}",
          "depth": 3,
          "lambda": "0",
          "formula": "m^4 + Z_3\u00b7(3^3, 0*3^3)",
          "size": 243
        },
        {
          "name": "I_{3,1}",
          "depth": 3,
          "lambda": "1",
          "formula": "m^4 + Z_3\u00b7(3^3, 1*3^3)",
          "size": 243
        },
        {
          "name": "I_{3,2}",
          "depth": 3,
          "lambda": "2",
          "formula": "m^4 + Z_3\u00b7(3^3, 2*3^3)",
          "size": 243
        },
        {
          "name": "I_{3,\u221e}",
          "depth": 3,
          "lambda": "inf",
          "formula": "m^4 + Z_3\u00b7(0, 3^3)",
          "size": 243
        },
        {
          "name": "I_{4,0}",
          "depth": 4,
          "lambda": "0",
          "formula": "m^5 + Z_3\u00b7(3^4, 0*3^4)",
          "size": 27
        },
        {
          "name": "I_{4,1}",
          "depth": 4,
          "lambda": "1",
          "formula": "m^5 + Z_3\u00b7(3^4, 1*3^4)",
          "size": 27
        },
        {
          "name": "I_{4,2}",
          "depth": 4,
          "lambda": "2",
          "formula": "m^5 + Z_3\u00b7(3^4, 2*3^4)",
          "size": 27
        },
        {
          "name": "I_{4,\u221e}",
          "depth": 4,
          "lambda": "inf",
          "formula": "m^5 + Z_3\u00b7(0, 3^4)",
          "size": 27
        },
        {
          "name": "I_{5,0}",
          "depth": 5,
          "lambda": "0",
          "formula": "m^6 + Z_3\u00b7(3^5, 0*3^5)",
          "size": 3
        },
        {
          "name": "I_{5,1}",
          "depth": 5,
          "lambda": "1",
          "formula": "m^6 + Z_3\u00b7(3^5, 1*3^5)",
          "size": 3
        },
        {
          "name": "I_{5,2}",
          "depth": 5,
          "lambda": "2",
          "formula": "m^6 + Z_3\u00b7(3^5, 2*3^5)",
          "size": 3
        },
        {
          "name": "I_{5,\u221e}",
          "depth": 5,
          "lambda": "inf",
          "formula": "m^6 + Z_3\u00b7(0, 3^5)",
          "size": 3
        }
      ],
      "ideal_count_formula": 27
    }
  }
}