{
  "arithmetic": "exact integers and fractions",
  "python": "3.12.11",
  "method": "enumerate every first row and all second rows; test determinant one",
  "all_assertions_passed": true,
  "prime_cases": [
    {
      "prime": 3,
      "group_order": 24,
      "zeros_a2_plus_b2": 0,
      "beta_a2_plus_b2": "0",
      "zeros_a2_plus_1": 0,
      "beta_a2_plus_1": "0"
    },
    {
      "prime": 5,
      "group_order": 120,
      "zeros_a2_plus_b2": 40,
      "beta_a2_plus_b2": "1/3",
      "zeros_a2_plus_1": 50,
      "beta_a2_plus_1": "5/12"
    },
    {
      "prime": 7,
      "group_order": 336,
      "zeros_a2_plus_b2": 0,
      "beta_a2_plus_b2": "0",
      "zeros_a2_plus_1": 0,
      "beta_a2_plus_1": "0"
    },
    {
      "prime": 11,
      "group_order": 1320,
      "zeros_a2_plus_b2": 0,
      "beta_a2_plus_b2": "0",
      "zeros_a2_plus_1": 0,
      "beta_a2_plus_1": "0"
    },
    {
      "prime": 13,
      "group_order": 2184,
      "zeros_a2_plus_b2": 312,
      "beta_a2_plus_b2": "1/7",
      "zeros_a2_plus_1": 338,
      "beta_a2_plus_1": "13/84"
    },
    {
      "prime": 17,
      "group_order": 4896,
      "zeros_a2_plus_b2": 544,
      "beta_a2_plus_b2": "1/9",
      "zeros_a2_plus_1": 578,
      "beta_a2_plus_1": "17/144"
    },
    {
      "prime": 19,
      "group_order": 6840,
      "zeros_a2_plus_b2": 0,
      "beta_a2_plus_b2": "0",
      "zeros_a2_plus_1": 0,
      "beta_a2_plus_1": "0"
    },
    {
      "prime": 23,
      "group_order": 12144,
      "zeros_a2_plus_b2": 0,
      "beta_a2_plus_b2": "0",
      "zeros_a2_plus_1": 0,
      "beta_a2_plus_1": "0"
    },
    {
      "prime": 29,
      "group_order": 24360,
      "zeros_a2_plus_b2": 1624,
      "beta_a2_plus_b2": "1/15",
      "zeros_a2_plus_1": 1682,
      "beta_a2_plus_1": "29/420"
    },
    {
      "prime": 31,
      "group_order": 29760,
      "zeros_a2_plus_b2": 0,
      "beta_a2_plus_b2": "0",
      "zeros_a2_plus_1": 0,
      "beta_a2_plus_1": "0"
    }
  ],
  "crt_ratios": [
    {
      "number_of_prime_factors": 1,
      "modulus": 5,
      "ratio_to_m2": "8/5"
    },
    {
      "number_of_prime_factors": 2,
      "modulus": 65,
      "ratio_to_m2": "192/65"
    },
    {
      "number_of_prime_factors": 3,
      "modulus": 1105,
      "ratio_to_m2": "6144/1105"
    },
    {
      "number_of_prime_factors": 4,
      "modulus": 32045,
      "ratio_to_m2": "344064/32045"
    },
    {
      "number_of_prime_factors": 5,
      "modulus": 1185665,
      "ratio_to_m2": "24772608/1185665"
    },
    {
      "number_of_prime_factors": 6,
      "modulus": 48612265,
      "ratio_to_m2": "396361728/9722453"
    },
    {
      "number_of_prime_factors": 7,
      "modulus": 2576450045,
      "ratio_to_m2": "3170893824/39637693"
    }
  ]
}