Sato-Tate group data - 1.6.B.1.1a

Formats: - HTML - YAML - JSON - 2024-04-25T06:20:41.372660

• 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, 2, 3, 4, 4, 8, 9, 9, 12, 10, 6, 17, 22, 26, 40, 24, 19, 40, 30, 20], 'character_matrix': [[1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 2, 0, 0, 2, 0, 2, 0, 0, 2, 0], [0, 0, 3, 0, 1, 0, 3, 1, 0, 0, 0, 4, 0, 0, 4], [0, 0, 0, 4, 0, 2, 0, 0, 0, 0, 0, 0, 6, 2, 0], [1, 0, 1, 0, 4, 0, 1, 4, 0, 3, 0, 4, 0, 0, 4], [0, 2, 0, 2, 0, 8, 0, 0, 6, 0, 4, 0, 8, 10, 0], [0, 0, 3, 0, 1, 0, 9, 3, 0, 0, 0, 8, 0, 0, 12], [1, 0, 1, 0, 4, 0, 3, 9, 0, 6, 0, 7, 0, 0, 12], [0, 2, 0, 0, 0, 6, 0, 0, 12, 0, 4, 0, 6, 12, 0], [1, 0, 0, 0, 3, 0, 0, 6, 0, 10, 0, 3, 0, 0, 8], [0, 2, 0, 0, 0, 4, 0, 0, 4, 0, 6, 0, 2, 8, 0], [0, 0, 4, 0, 4, 0, 8, 7, 0, 3, 0, 17, 0, 0, 20], [0, 0, 0, 6, 0, 8, 0, 0, 6, 0, 2, 0, 22, 14, 0], [0, 2, 0, 2, 0, 10, 0, 0, 12, 0, 8, 0, 14, 26, 0], [0, 0, 4, 0, 4, 0, 12, 12, 0, 8, 0, 20, 0, 0, 40]], 'component_group': '1.1', 'component_group_number': 1, 'components': 1, 'counts': [], 'degree': 6, 'first_a2_moment': 1, 'fourth_trace_moment': 12, 'gens': '', 'identity_component': 'U(3)', 'label': '1.6.B.1.1a', 'label_components': [1, 6, 1, 1, 1, 0], 'maximal': False, 'moments': [['a_1', 1, 0, 2, 0, 12, 0, 120, 0, 1610, 0, 25956, 0, 474012], ['a_2', 1, 1, 4, 18, 107, 743, 5793, 49157, 445133, 4243629, 42183426, 434117676, 4600195119], ['a_3', 1, 0, 6, 0, 232, 0, 21260, 0, 2844856, 0, 472811472, 0, 90764342592]], 'name': 'U(3)', 'pretty': '\\mathrm{U}(3)', 'rational': True, 'real_dimension': 9, 'second_trace_moment': 2, 'simplex': [1, 2, 4, 2, 6, 12, 6, 18, 12, 32, 20, 60, 120, 24, 107, 68, 46, 202, 130, 396, 250, 790, 1610, 154, 743, 96, 466, 294, 1470, 920, 584, 2964, 1850, 6052, 3752, 12474, 25956, 232, 1108, 5793, 696, 3570, 2214, 11816, 1374, 7262, 4482, 24378, 14928, 9194, 50712, 30954, 106218, 64596, 223776, 474012], 'st0_label': '1.6.B', 'subgroup_multiplicities': [], 'subgroups': [], 'supgroup_multiplicities': [1], 'supgroups': ['1.6.B.2.1a'], 'trace_histogram': 'data:image/png;base64,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', 'trace_zero_density': '0', 'weight': 1, 'zvector': [0, 0, 0, 0, 0, 0, 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&\\overline{A}\\end{bmatrix}: A\\in\\mathrm{U}(3)\\right\\}', 'hodge_circle': 'u\\mapsto\\mathrm{diag}(u,u,u,\\bar u,\\bar u,\\bar u)', 'label': '1.6.B', 'label_components': [1, 6, 1], 'name': 'U(3)', 'pretty': '\\mathrm{U}(3)', 'real_dimension': 9, 'symplectic_form': '\\begin{bmatrix}0&I_3\\\\-I_3&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': '1.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_gens': [], 'aut_group': '1.1', 'aut_order': 1, 'aut_stats': [[1, 1, 1, 1]], 'cc_stats': [[1, 1, 1]], 'center_label': '1.1', 'central_product': False, 'central_quotient': '1.1', 'commutator_count': 0, 'commutator_label': '1.1', 'complements_known': True, 'complete': True, 'complex_characters_known': True, 'composition_factors': [], 'composition_length': 0, 'counter': 1, 'cyclic': True, 'derived_length': 0, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 1, 'exponent': 1, 'exponents_of_order': [], 'factors_of_aut_order': [], 'factors_of_order': [], 'faithful_reps': [[1, 1, 1]], 'frattini_label': '1.1', 'frattini_quotient': '1.1', 'hash': 1, 'hyperelementary': 1, 'irrC_degree': 1, 'irrQ_degree': 1, 'irrep_stats': [[1, 1]], 'label': '1.1', 'linC_degree': 0, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': None, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C1', 'ngens': 0, 'nilpotency_class': 0, 'nilpotent': True, 'normal_counts': None, 'normal_index_bound': None, 'normal_order_bound': None, 'normal_subgroups_known': True, 'number_autjugacy_classes': 1, 'number_characteristic_subgroups': 1, 'number_conjugacy_classes': 1, 'number_divisions': 1, 'number_normal_subgroups': 1, 'number_subgroup_autclasses': 1, 'number_subgroup_classes': 1, 'number_subgroups': 1, 'old_label': None, 'order': 1, 'order_factorization_type': 0, 'order_stats': [[1, 1]], 'outer_equivalence': False, 'outer_group': '1.1', 'outer_order': 1, 'pc_rank': 0, 'perfect': True, 'permutation_degree': 1, 'pgroup': 1, 'primary_abelian_invariants': [], 'quasisimple': False, 'rank': 0, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 1]], 'representations': {'PC': {'code': 0, 'gens': []}, 'Perm': {'d': 1, 'gens': []}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_1', 'transitive_degree': 1, 'wreath_data': None, 'wreath_product': False}