{
  "schema": "cyclotomic-real-class-numbers/v1",
  "finite_certificate": {
    "coefficient_convention": "All polynomial coefficient lists are ascending. Basis coefficients use b_0=1 and b_j=S_j(t)=zeta_p^j+zeta_p^(-j), 1<=j<n. Matrices have row i and column j equal to coefficient of T^i in g(T)T^j mod F_p(T).",
    "method": "Unconditional Minkowski bound, complete small-prime-orbit enumeration, exact integer norm determinants. No GRH, class-field analytic bounds, precomputed class group, paid proof or unit-regulator computation.",
    "fields": [
      {
        "conductor": 3,
        "degree": 1,
        "minimal_polynomial_coefficients_ascending": [
          1,
          1
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        "discriminant": 1,
        "minkowski_bound_floor": 1,
        "minkowski_squared_numerator": 1,
        "minkowski_squared_denominator": 1,
        "minkowski_bound_approx_display_only": 1.0,
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        "principal_generators": []
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      {
        "conductor": 5,
        "degree": 2,
        "minimal_polynomial_coefficients_ascending": [
          -1,
          1,
          1
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        "discriminant": 5,
        "minkowski_bound_floor": 1,
        "minkowski_squared_numerator": 20,
        "minkowski_squared_denominator": 16,
        "minkowski_bound_approx_display_only": 1.118033988749895,
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        "principal_generators": []
      },
      {
        "conductor": 7,
        "degree": 3,
        "minimal_polynomial_coefficients_ascending": [
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          -2,
          1,
          1
        ],
        "discriminant": 49,
        "minkowski_bound_floor": 1,
        "minkowski_squared_numerator": 1764,
        "minkowski_squared_denominator": 729,
        "minkowski_bound_approx_display_only": 1.5555555555555556,
        "required_prime_orbits": [],
        "principal_generators": []
      },
      {
        "conductor": 11,
        "degree": 5,
        "minimal_polynomial_coefficients_ascending": [
          1,
          3,
          -3,
          -4,
          1,
          1
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        "discriminant": 14641,
        "minkowski_bound_floor": 4,
        "minkowski_squared_numerator": 210830400,
        "minkowski_squared_denominator": 9765625,
        "minkowski_bound_approx_display_only": 4.6464,
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        "principal_generators": []
      },
      {
        "conductor": 13,
        "degree": 6,
        "minimal_polynomial_coefficients_ascending": [
          -1,
          3,
          6,
          -4,
          -5,
          1,
          1
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        "discriminant": 371293,
        "minkowski_bound_floor": 9,
        "minkowski_squared_numerator": 192478291200,
        "minkowski_squared_denominator": 2176782336,
        "minkowski_bound_approx_display_only": 9.403366752367504,
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        "principal_generators": []
      },
      {
        "conductor": 17,
        "degree": 8,
        "minimal_polynomial_coefficients_ascending": [
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          -10,
          10,
          15,
          -6,
          -7,
          1,
          1
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        "discriminant": 410338673,
        "minkowski_bound_floor": 48,
        "minkowski_squared_numerator": 667088565508915200,
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            "residue_degree": 4,
            "prime_ideal_norm": 16,
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            "ramification_index": 1
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          {
            "rational_prime": 17,
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        "principal_generators": [
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            "signed_norm": 16,
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              0,
              0,
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              -1
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            "discovery_support": 3,
            "origin": "New sparse-vector discovery; exact norm verified by integer resultant and independently by integer multiplication determinant.",
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            "generator_basis_display": "1 -b_1 -b_4"
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          {
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            "origin": "Explicit ramified-prime generator; N(2-t)=F_p(2)=p.",
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            "generator_basis_display": "2 -b_1"
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      {
        "conductor": 19,
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        "discriminant": 16983563041,
        "minkowski_bound_floor": 122,
        "minkowski_squared_numerator": 2236427754900704870400,
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            "prime_ideal_norm": 19,
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            "ramification_index": 9
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          {
            "rational_prime": 37,
            "residue_degree": 1,
            "prime_ideal_norm": 37,
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            "ramification_index": 1
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          {
            "rational_prime": 113,
            "residue_degree": 1,
            "prime_ideal_norm": 113,
            "number_of_conjugate_primes": 9,
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        ],
        "principal_generators": [
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            "absolute_norm": 19,
            "signed_norm": 19,
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            "g": "2 - T",
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            "origin": "Explicit ramified-prime generator; N(2-t)=F_p(2)=p.",
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            "generator_basis_display": "2 -b_1"
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          {
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            "g": "T**3 - 4*T + 1",
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            "discovery_support": 3,
            "origin": "New sparse-vector discovery; exact norm verified by integer resultant and independently by integer multiplication determinant.",
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          {
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            "g": "-T**2 + T + 3",
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            "origin": "New sparse-vector discovery; exact norm verified by integer resultant and independently by integer multiplication determinant.",
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        "discriminant": 41426511213649,
        "minkowski_bound_floor": 900,
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          {
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          {
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          {
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          {
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          {
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          {
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          {
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        "principal_generators": [
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