Sato-Tate group data - 1.2.A.1.1a

Formats: - HTML - YAML - JSON - 2025-03-25T22:05:21.180538

• 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[]

  
  {'component_group': '1.1', 'component_group_number': 1, 'components': 1, 'counts': [], 'degree': 2, 'fourth_trace_moment': 2, 'gens': [], 'identity_component': 'SU(2)', 'label': '1.2.A.1.1a', 'label_components': [1, 2, 0, 1, 1, 0], 'maximal': True, 'moments': [['a_1', 1, 0, 1, 0, 2, 0, 5, 0, 14, 0, 42, 0, 132]], 'name': 'SU(2)', 'old_label': '1.2.3.1.1a', 'pretty': '\\mathrm{SU}(2)', 'rational': True, 'real_dimension': 3, 'second_trace_moment': 1, 'st0_label': '1.2.A', 'subgroup_multiplicities': [], 'subgroups': [], 'supgroups': [], 'trace_histogram': 'data:image/png;base64,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', 'trace_zero_density': '0', 'weight': 1, 'zvector': [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': 2, 'description': '\\left\\{\\begin{bmatrix}\\alpha&\\beta\\\\-\\bar\\beta&\\bar\\alpha\\end{bmatrix}:\\alpha\\bar\\alpha+\\beta\\bar\\beta = 1,\\ \\alpha,\\beta\\in\\mathbb{C}\\right\\}', 'hodge_circle': 'u\\mapsto\\mathrm{diag}(u,\\bar u)', 'label': '1.2.A', 'label_components': [1, 2, 0], 'name': 'SU(2)', 'pretty': '\\mathrm{SU}(2)', 'real_dimension': 3, 'symplectic_form': '\\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
  irrQ_dim	integer
  irrR_degree	integer
  irrep_stats	numeric[]
  label	text
  linC_count	numeric
  linC_degree	integer
  linFp_degree	integer
  linFq_degree	integer
  linQ_degree	integer
  linQ_degree_count	numeric
  linQ_dim	integer
  linQ_dim_count	numeric
  linR_count	numeric
  linR_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, 'irrQ_dim': 1, 'irrR_degree': 1, 'irrep_stats': [[1, 1]], 'label': '1.1', 'linC_count': 1, 'linC_degree': 0, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 0, 'linQ_degree_count': 1, 'linQ_dim': 0, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 0, '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': [], 'pres': []}, '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}