Sato-Tate group data - 1.4.F.12.4d

Formats: - HTML - YAML - JSON - 2024-04-25T07:28:14.410816

• 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, 8, 12, 10, 14, 37, 40, 23], 'character_matrix': [[1, 0, 1, 0, 0, 2, 0, 1, 0, 2], [0, 2, 0, 0, 4, 0, 4, 0, 6, 0], [1, 0, 3, 2, 0, 4, 0, 7, 0, 6], [0, 0, 2, 8, 0, 4, 0, 14, 0, 6], [0, 4, 0, 0, 12, 0, 12, 0, 20, 0], [2, 0, 4, 4, 0, 10, 0, 14, 0, 12], [0, 4, 0, 0, 12, 0, 14, 0, 22, 0], [1, 0, 7, 14, 0, 14, 0, 37, 0, 24], [0, 6, 0, 0, 20, 0, 22, 0, 40, 0], [2, 0, 6, 6, 0, 12, 0, 24, 0, 23]], 'component_group': '12.4', 'component_group_number': 4, 'components': 12, 'counts': [['a_1', [[0, 7]]], ['a_2', [[2, 6]]]], 'degree': 4, 'first_a2_moment': 2, 'fourth_trace_moment': 18, 'gens': [[['\\zeta_{12}', '0', '0', '0'], ['0', '\\zeta_{12}^{11}', '0', '0'], ['0', '0', '\\zeta_{12}^{11}', '0'], ['0', '0', '0', '\\zeta_{12}']], [['0', '0', '-1', '0'], ['0', '0', '0', '-1'], ['1', '0', '0', '0'], ['0', '1', '0', '0']]], 'identity_component': 'U(1)_2', 'label': '1.4.F.12.4d', 'label_components': [1, 4, 5, 12, 4, 3], 'maximal': False, 'moments': [['a_1', 1, 0, 2, 0, 18, 0, 200, 0, 2450, 0, 31752, 0, 427812], ['a_2', 1, 2, 6, 20, 82, 372, 1825, 9326, 49026, 262628, 1427641, 7851846, 43609857]], 'name': 'D_{6,2}', 'old_label': '1.4.1.12.4d', 'pretty': 'D_{6,2}', 'rational': True, 'real_dimension': 1, 'second_trace_moment': 2, 'simplex': [2, 2, 6, 8, 18, 20, 36, 84, 200, 82, 172, 412, 1000, 2450, 372, 856, 2088, 5140, 12740, 31752, 1825, 4386, 10842, 26980, 67494, 169596, 427812], 'st0_label': '1.4.F', 'subgroup_multiplicities': [1, 2, 1], 'subgroups': ['1.4.F.4.2c', '1.4.F.6.1b', '1.4.F.6.2c'], 'supgroups': ['1.4.F.24.14a'], 'trace_histogram': 'data:image/png;base64,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', 'trace_zero_density': '7/12', 'weight': 1, 'zvector': [7, 0, 0, 0, 0, 6, 0, 0, 0, 0, 6]}
• 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': False, 'abelian': False, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_gens': [[1, 2], [7, 10], [1, 10], [5, 2]], 'aut_group': '12.4', 'aut_order': 12, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 2, 1], [3, 2, 1, 1], [6, 2, 1, 1]], 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 3, 2], [3, 2, 1], [6, 2, 1]], 'center_label': '2.1', 'central_product': True, 'central_quotient': '6.1', 'commutator_count': 1, 'commutator_label': '3.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '3.1'], 'composition_length': 3, 'counter': 4, 'cyclic': False, 'derived_length': 2, 'direct_factorization': [['2.1', 1], ['6.1', 1]], 'direct_product': True, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 3, 1, 2], [3, 2, 1, 1], [6, 2, 1, 1]], 'element_repr_type': 'PC', 'elementary': 1, 'eulerian_function': 3, 'exponent': 6, 'exponents_of_order': [2, 1], 'factors_of_aut_order': [2, 3], 'factors_of_order': [2, 3], 'faithful_reps': [[2, 1, 1]], 'frattini_label': '1.1', 'frattini_quotient': '12.4', 'hash': 4, 'hyperelementary': 2, 'irrC_degree': 2, 'irrQ_degree': 2, 'irrep_stats': [[1, 4], [2, 2]], 'label': '12.4', 'linC_degree': None, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D6', 'ngens': 3, 'nilpotency_class': -1, 'nilpotent': False, 'normal_counts': None, 'normal_index_bound': None, 'normal_order_bound': None, 'normal_subgroups_known': True, 'number_autjugacy_classes': 5, 'number_characteristic_subgroups': 5, 'number_conjugacy_classes': 6, 'number_divisions': 6, 'number_normal_subgroups': 7, 'number_subgroup_autclasses': 8, 'number_subgroup_classes': 10, 'number_subgroups': 16, 'old_label': None, 'order': 12, 'order_factorization_type': 22, 'order_stats': [[1, 1], [2, 7], [3, 2], [6, 2]], 'outer_equivalence': False, 'outer_group': '2.1', 'outer_order': 2, 'pc_rank': 2, 'perfect': False, 'permutation_degree': 5, 'pgroup': 0, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 2]], 'representations': {'PC': {'code': 43377, 'gens': [1, 2], 'pres': [3, -2, -2, -3, 61, 16, 74]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [17, 35]}, 'GLFp': {'d': 2, 'p': 3, 'gens': [31, 55, 56]}, 'Perm': {'d': 5, 'gens': [6, 1, 30]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 6, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_6', 'transitive_degree': 6, 'wreath_data': None, 'wreath_product': False}