Sato-Tate group data - 1.4.F.4.1b

Formats: - HTML - YAML - JSON - 2024-04-18T16:49:23.872470

• 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, 4, 5, 12, 24, 18, 28, 79, 96, 54], 'character_matrix': [[1, 0, 1, 2, 0, 2, 0, 3, 0, 2], [0, 4, 0, 0, 8, 0, 8, 0, 12, 0], [1, 0, 5, 6, 0, 6, 0, 15, 0, 12], [2, 0, 6, 12, 0, 12, 0, 26, 0, 18], [0, 8, 0, 0, 24, 0, 24, 0, 44, 0], [2, 0, 6, 12, 0, 18, 0, 32, 0, 24], [0, 8, 0, 0, 24, 0, 28, 0, 48, 0], [3, 0, 15, 26, 0, 32, 0, 79, 0, 62], [0, 12, 0, 0, 44, 0, 48, 0, 96, 0], [2, 0, 12, 18, 0, 24, 0, 62, 0, 54]], 'component_group': '4.1', 'component_group_number': 1, 'components': 4, 'counts': [['a_1', [[0, 1]]]], 'degree': 4, 'first_a2_moment': 2, 'fourth_trace_moment': 36, 'gens': [[['\\zeta_8', '0', '0', '0'], ['0', '\\zeta_8^7', '0', '0'], ['0', '0', '\\zeta_8^7', '0'], ['0', '0', '0', '\\zeta_8']]], 'identity_component': 'U(1)_2', 'label': '1.4.F.4.1b', 'label_components': [1, 4, 5, 4, 1, 1], 'maximal': False, 'moments': [['a_1', 1, 0, 4, 0, 36, 0, 400, 0, 5040, 0, 68544, 0, 975744], ['a_2', 1, 2, 8, 32, 150, 732, 3776, 20064, 109318, 605804, 3400848, 19273344, 110017980]], 'name': 'C_4', 'old_label': '1.4.1.4.1a', 'pretty': 'C_4', 'rational': True, 'real_dimension': 1, 'second_trace_moment': 4, 'simplex': [2, 4, 8, 16, 36, 32, 72, 168, 400, 150, 348, 836, 2040, 5040, 732, 1768, 4344, 10800, 27104, 68544, 3776, 9312, 23272, 58672, 148960, 380352, 975744], 'st0_label': '1.4.F', 'subgroup_multiplicities': [1], 'subgroups': ['1.4.F.2.1b'], 'supgroups': ['1.4.F.8.2a', '1.4.F.8.3b', '1.4.F.8.3c'], 'trace_histogram': 'data:image/png;base64,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', 'trace_zero_density': '1/4', 'weight': 1, 'zvector': [1, 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': 4, 'description': '\\left\\{\\begin{bmatrix}\\alpha I_2&0\\\\0&\\bar\\alpha I_2\\end{bmatrix}: \\alpha\\bar\\alpha = 1,\\ \\alpha\\in\\mathbb{C}\\right\\}', 'hodge_circle': 'u\\mapsto\\mathrm{diag}(u, u,\\bar u,\\bar u)', 'label': '1.4.F', 'label_components': [1, 4, 5], 'name': 'U(1)_2', 'pretty': '\\mathrm{U}(1)_2', 'real_dimension': 1, 'symplectic_form': '\\begin{bmatrix}0&I_2\\\\-I_2&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': True, 'abelian': True, 'abelian_quotient': '4.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_gens': [[1], [3]], 'aut_group': '2.1', 'aut_order': 2, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1]], 'cc_stats': [[1, 1, 1], [2, 1, 1], [4, 1, 2]], 'center_label': '4.1', 'central_product': False, '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': 1, 'cyclic': True, 'derived_length': 1, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 4, 'exponents_of_order': [2], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[1, 0, 2]], 'frattini_label': '2.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, 'irrC_degree': 1, 'irrQ_degree': 2, 'irrep_stats': [[1, 4]], 'label': '4.1', 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C4', 'ngens': 1, 'nilpotency_class': 1, 'nilpotent': True, 'normal_counts': None, 'normal_index_bound': None, 'normal_order_bound': None, 'normal_subgroups_known': True, 'number_autjugacy_classes': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 4, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 3, 'number_subgroups': 3, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 1], [4, 2]], 'outer_equivalence': False, 'outer_group': '2.1', 'outer_order': 2, 'pc_rank': 1, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1]], 'representations': {'PC': {'code': 5, 'gens': [1], 'pres': [2, -2, -2, 4]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [46]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [56, 15], 'family': 'CSOPlus'}], 'GLFp': {'d': 2, 'p': 3, 'gens': [21]}, 'Perm': {'d': 4, 'gens': [22, 7]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4', 'transitive_degree': 4, 'wreath_data': None, 'wreath_product': False}