Sato-Tate group data - 1.6.K.4.2b

Formats: - HTML - YAML - JSON - 2024-05-03T11:56:00.174168

• gps_st •

  Column	Type
  character_diagonal	bigint[]
  character_matrix	bigint[]
  component_group	text
  component_group_number	smallint
  components	smallint
  counts	jsonb
  degree	smallint
  first_a2_moment	smallint
  fourth_trace_moment	smallint
  gens	jsonb
  identity_component	text
  label	text
  label_components	integer[]
  maximal	boolean
  moments	jsonb
  name	text
  old_label	text
  pretty	text
  rational	boolean
  real_dimension	smallint
  second_trace_moment	smallint
  simplex	bigint[]
  st0_label	text
  subgroup_multiplicities	smallint[]
  subgroups	jsonb
  supgroup_multiplicities	smallint[]
  supgroups	jsonb
  trace_histogram	text
  trace_zero_density	text
  weight	smallint
  zvector	smallint[]

  
  {'character_diagonal': [1, 3, 9, 8, 14, 60, 61, 88, 129, 57, 40, 280, 312, 654, 1192, 546, 577, 1265, 987, 279], 'character_matrix': [[1, 0, 1, 0, 1, 0, 1, 5, 0, 1, 0, 2, 0, 0, 6], [0, 3, 0, 2, 0, 10, 0, 0, 13, 0, 7, 0, 14, 27, 0], [1, 0, 9, 0, 5, 0, 17, 20, 0, 9, 0, 35, 0, 0, 64], [0, 2, 0, 8, 0, 16, 0, 0, 17, 0, 8, 0, 37, 46, 0], [1, 0, 5, 0, 14, 0, 21, 25, 0, 21, 0, 49, 0, 0, 86], [0, 10, 0, 16, 0, 60, 0, 0, 80, 0, 44, 0, 124, 184, 0], [1, 0, 17, 0, 21, 0, 61, 59, 0, 40, 0, 122, 0, 0, 242], [5, 0, 20, 0, 25, 0, 59, 88, 0, 55, 0, 132, 0, 0, 284], [0, 13, 0, 17, 0, 80, 0, 0, 129, 0, 60, 0, 176, 277, 0], [1, 0, 9, 0, 21, 0, 40, 55, 0, 57, 0, 99, 0, 0, 220], [0, 7, 0, 8, 0, 44, 0, 0, 60, 0, 40, 0, 91, 140, 0], [2, 0, 35, 0, 49, 0, 122, 132, 0, 99, 0, 280, 0, 0, 544], [0, 14, 0, 37, 0, 124, 0, 0, 176, 0, 91, 0, 312, 428, 0], [0, 27, 0, 46, 0, 184, 0, 0, 277, 0, 140, 0, 428, 654, 0], [6, 0, 64, 0, 86, 0, 242, 284, 0, 220, 0, 544, 0, 0, 1192]], 'component_group': '4.2', 'component_group_number': 2, 'components': 4, 'counts': [['a_2', [[-1, 1], [3, 1]]]], 'degree': 6, 'first_a2_moment': 2, 'fourth_trace_moment': 38, 'gens': '\\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 1 & 0 & 0 & 0 & 0 \\\\0 & 0 & i & 0 &0 & 0 \\\\0 & 0 & 0 & -i & 0 & 0 \\\\0 & 0 & 0 & 0 & -i & 0 \\\\0 & 0 & 0 & 0 & 0 & i \\\\\\end{bmatrix}, \\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 1 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 1 \\\\0 & 0 & 0 & 0 & -1 & 0 \\\\0 & 0 & 0 & -1 & 0 & 0 \\\\0 & 0 & 1 & 0 & 0 & 0 \\\\\\end{bmatrix}', 'identity_component': 'SU(2)xU(1)_2', 'label': '1.6.K.4.2b', 'label_components': [1, 6, 10, 4, 2, 1], 'maximal': False, 'moments': [['a_1', 1, 0, 3, 0, 38, 0, 745, 0, 17094, 0, 427014, 0, 11289564], ['a_2', 1, 2, 12, 83, 750, 7697, 84989, 981766, 11707026, 143022917, 1781477919, 22546777596, 289206317739], ['a_3', 1, 0, 15, 0, 1878, 0, 386170, 0, 94401258, 0, 25461220092, 0, 7330330362168]], 'name': 'SU(2)xJ(C_2)', 'pretty': '\\mathrm{SU}(2)\\times J(C_2)', 'rational': True, 'real_dimension': 4, 'second_trace_moment': 3, 'simplex': [2, 3, 12, 5, 18, 38, 15, 83, 46, 160, 92, 340, 745, 117, 750, 425, 250, 1581, 918, 3466, 2005, 7670, 17094, 1166, 7697, 668, 4399, 2532, 16904, 9704, 5595, 37646, 21590, 84290, 48244, 189420, 427014, 1878, 12444, 84989, 7138, 48375, 27636, 190177, 15810, 108264, 61753, 428092, 243353, 138630, 966747, 548786, 2188760, 1240722, 4966164, 11289564], 'st0_label': '1.6.K', 'subgroup_multiplicities': [1, 1, 1], 'subgroups': ['1.6.K.2.1c', '1.6.K.2.1a', '1.6.K.2.1b'], 'supgroup_multiplicities': [3, 1, 1, 1], 'supgroups': ['1.6.K.8.5a', '1.6.K.12.4a', '1.6.K.8.2a', '1.6.K.12.5a'], 'trace_histogram': 'data:image/png;base64,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', 'trace_zero_density': '0', 'weight': 1, 'zvector': [0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]}
• gps_st0 •

  Column	Type
  degree	smallint
  description	text
  hodge_circle	text
  label	text
  label_components	integer[]
  name	text
  pretty	text
  real_dimension	smallint
  symplectic_form	text
  weight	smallint

  
  {'degree': 6, 'description': '\\left\\{\\begin{bmatrix}A&0&0\\\\0&\\alpha I_2&0\\\\0&0&\\bar\\alpha I_2\\end{bmatrix}: A\\in\\mathrm{SU}(2),\\ \\alpha\\bar\\alpha = 1,\\ \\alpha\\in\\mathbb{C}\\right\\}', 'hodge_circle': 'u\\mapsto\\mathrm{diag}(u,\\bar u, u, u,\\bar u, \\bar u)', 'label': '1.6.K', 'label_components': [1, 6, 10], 'name': 'SU(2)xU(1)_2', 'pretty': '\\mathrm{SU}(2)\\times\\mathrm{U}(1)_2', 'real_dimension': 4, 'symplectic_form': '\\begin{bmatrix}J_2&0&0\\\\0&0&I_2\\\\0&-I_2&0\\end{bmatrix},\\ J_2:=\\begin{bmatrix}0&1\\\\-1&0\\end{bmatrix}', 'weight': 1}
• gps_groups •

  Column	Type
  Agroup	boolean
  Zgroup	boolean
  abelian	boolean
  abelian_quotient	text
  all_subgroups_known	boolean
  almost_simple	boolean
  aut_gens	numeric[]
  aut_group	text
  aut_order	numeric
  aut_stats	numeric[]
  cc_stats	numeric[]
  center_label	text
  central_product	boolean
  central_quotient	text
  commutator_count	integer
  commutator_label	text
  complements_known	boolean
  complete	boolean
  complex_characters_known	boolean
  composition_factors	text[]
  composition_length	smallint
  counter	integer
  cyclic	boolean
  derived_length	smallint
  direct_factorization	jsonb
  direct_product	boolean
  div_stats	numeric[]
  element_repr_type	text
  elementary	integer
  eulerian_function	numeric
  exponent	numeric
  exponents_of_order	smallint[]
  factors_of_aut_order	integer[]
  factors_of_order	smallint[]
  faithful_reps	numeric[]
  frattini_label	text
  frattini_quotient	text
  hash	bigint
  hyperelementary	integer
  irrC_degree	integer
  irrQ_degree	integer
  irrep_stats	numeric[]
  label	text
  linC_degree	integer
  linFp_degree	integer
  linFq_degree	integer
  linQ_degree	integer
  maximal_subgroups_known	boolean
  metabelian	boolean
  metacyclic	boolean
  monomial	boolean
  name	text
  ngens	smallint
  nilpotency_class	smallint
  nilpotent	boolean
  normal_counts	integer[]
  normal_index_bound	numeric
  normal_order_bound	numeric
  normal_subgroups_known	boolean
  number_autjugacy_classes	integer
  number_characteristic_subgroups	integer
  number_conjugacy_classes	integer
  number_divisions	integer
  number_normal_subgroups	integer
  number_subgroup_autclasses	integer
  number_subgroup_classes	integer
  number_subgroups	numeric
  old_label	text
  order	numeric
  order_factorization_type	integer
  order_stats	numeric[]
  outer_equivalence	boolean
  outer_group	text
  outer_order	numeric
  pc_rank	smallint
  perfect	boolean
  permutation_degree	integer
  pgroup	smallint
  primary_abelian_invariants	integer[]
  quasisimple	boolean
  rank	smallint
  rational	boolean
  rational_characters_known	boolean
  ratrep_stats	numeric[]
  representations	jsonb
  schur_multiplier	integer[]
  semidirect_product	boolean
  simple	boolean
  smith_abelian_invariants	integer[]
  solvability_type	smallint
  solvable	boolean
  subgroup_inclusions_known	boolean
  subgroup_index_bound	numeric
  supersolvable	boolean
  sylow_subgroups_known	boolean
  tex_name	text
  transitive_degree	integer
  wreath_data	text[]
  wreath_product	boolean

  
  {'Agroup': True, 'Zgroup': False, 'abelian': True, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_gens': [[1, 2], [3, 2], [2, 1]], 'aut_group': '6.1', 'aut_order': 6, 'aut_stats': [[1, 1, 1, 1], [2, 1, 3, 1]], 'cc_stats': [[1, 1, 1], [2, 1, 3]], 'center_label': '4.2', 'central_product': True, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1'], 'composition_length': 2, 'counter': 2, 'cyclic': False, 'derived_length': 1, 'direct_factorization': [['2.1', 2]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 3]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [2], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2], 'faithful_reps': [], 'frattini_label': '1.1', 'frattini_quotient': '4.2', 'hash': 2, 'hyperelementary': 2, 'irrC_degree': -1, 'irrQ_degree': -1, 'irrep_stats': [[1, 4]], 'label': '4.2', 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2^2', 'ngens': 2, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': None, 'normal_index_bound': None, 'normal_order_bound': None, 'normal_subgroups_known': True, 'number_autjugacy_classes': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 4, 'number_divisions': 4, 'number_normal_subgroups': 5, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 5, 'number_subgroups': 5, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 3]], 'outer_equivalence': False, 'outer_group': '6.1', 'outer_order': 6, 'pc_rank': 2, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4]], 'representations': {'PC': {'code': 0, 'gens': [1, 2], 'pres': [2, -2, 2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 12]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [55, 56]}, 'Perm': {'d': 4, 'gens': [6, 1]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 1, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2^2', 'transitive_degree': 4, 'wreath_data': None, 'wreath_product': False}