Sato-Tate group data - 1.4.F.2.1b

Formats: - HTML - YAML - JSON - 2024-05-01T05:43:24.415404

• 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, 7, 18, 40, 32, 52, 153, 184, 104], 'character_matrix': [[1, 0, 1, 2, 0, 4, 0, 5, 0, 4], [0, 4, 0, 0, 12, 0, 12, 0, 24, 0], [1, 0, 7, 8, 0, 10, 0, 27, 0, 24], [2, 0, 8, 18, 0, 22, 0, 48, 0, 36], [0, 12, 0, 0, 40, 0, 44, 0, 84, 0], [4, 0, 10, 22, 0, 32, 0, 60, 0, 46], [0, 12, 0, 0, 44, 0, 52, 0, 96, 0], [5, 0, 27, 48, 0, 60, 0, 153, 0, 122], [0, 24, 0, 0, 84, 0, 96, 0, 184, 0], [4, 0, 24, 36, 0, 46, 0, 122, 0, 104]], 'component_group': '2.1', 'component_group_number': 1, 'components': 2, 'counts': [['a_1', [[0, 1]]]], 'degree': 4, 'first_a2_moment': 2, 'fourth_trace_moment': 48, 'gens': [[['i', '0', '0', '0'], ['0', '-i', '0', '0'], ['0', '0', '-i', '0'], ['0', '0', '0', 'i']]], 'identity_component': 'U(1)_2', 'label': '1.4.F.2.1b', 'label_components': [1, 4, 5, 2, 1, 1], 'maximal': False, 'moments': [['a_1', 1, 0, 4, 0, 48, 0, 640, 0, 8960, 0, 129024, 0, 1892352], ['a_2', 1, 2, 10, 44, 230, 1212, 6628, 36696, 205766, 1162988, 6616940, 37841256, 217331804]], 'name': 'C_2', 'old_label': '1.4.1.2.1a', 'pretty': 'C_2', 'rational': True, 'real_dimension': 1, 'second_trace_moment': 4, 'simplex': [2, 4, 10, 20, 48, 44, 104, 256, 640, 230, 556, 1392, 3520, 8960, 1212, 3032, 7680, 19584, 50176, 129024, 6628, 16776, 42848, 109952, 283136, 731136, 1892352], 'st0_label': '1.4.F', 'subgroup_multiplicities': [1], 'subgroups': ['1.4.F.1.1a'], 'supgroups': ['1.4.F.4.1b', '1.4.F.4.1a', '1.4.F.4.2a', '1.4.F.4.2b', '1.4.F.4.2c', '1.4.F.6.1a', '1.4.F.6.2c'], 'trace_histogram': 'data:image/png;base64,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', 'trace_zero_density': '1/2', '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': '2.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_gens': [[1]], 'aut_group': '1.1', 'aut_order': 1, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'cc_stats': [[1, 1, 1], [2, 1, 1]], 'center_label': '2.1', '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'], 'composition_length': 1, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 2, 'exponents_of_order': [1], 'factors_of_aut_order': [], 'factors_of_order': [2], 'faithful_reps': [[1, 1, 1]], 'frattini_label': '1.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, 'irrC_degree': 1, 'irrQ_degree': 1, 'irrep_stats': [[1, 2]], 'label': '2.1', 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C2', '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': 2, 'number_characteristic_subgroups': 2, 'number_conjugacy_classes': 2, 'number_divisions': 2, 'number_normal_subgroups': 2, 'number_subgroup_autclasses': 2, 'number_subgroup_classes': 2, 'number_subgroups': 2, 'old_label': None, 'order': 2, 'order_factorization_type': 1, 'order_stats': [[1, 1], [2, 1]], 'outer_equivalence': False, 'outer_group': '1.1', 'outer_order': 1, 'pc_rank': 1, 'perfect': False, 'permutation_degree': 2, 'pgroup': 2, 'primary_abelian_invariants': [2], 'quasisimple': False, 'rank': 1, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 2]], 'representations': {'PC': {'code': 0, 'gens': [1], 'pres': [1, -2]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [12]}, 'Lie': [{'d': 2, 'q': 2, 'gens': [6], 'family': 'CSOPlus'}, {'d': 2, 'q': 2, 'gens': [6], 'family': 'COPlus'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'AGammaL'}, {'d': 1, 'q': 2, 'gens': [1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 2, 'gens': [11]}, 'Perm': {'d': 2, 'gens': [1]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': True, 'smith_abelian_invariants': [2], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_2', 'transitive_degree': 2, 'wreath_data': None, 'wreath_product': False}