Sato-Tate group data - 1.4.E.8.3a

Formats: - HTML - YAML - JSON - 2026-03-04T20:17:17.669812

• 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, 1, 2, 3, 3, 5, 3, 8, 8, 9], 'character_matrix': [[1, 0, 0, 0, 0, 1, 0, 0, 0, 0], [0, 1, 0, 0, 1, 0, 1, 0, 1, 0], [0, 0, 2, 0, 0, 0, 0, 2, 0, 3], [0, 0, 0, 3, 0, 1, 0, 2, 0, 0], [0, 1, 0, 0, 3, 0, 2, 0, 4, 0], [1, 0, 0, 1, 0, 5, 0, 1, 0, 0], [0, 1, 0, 0, 2, 0, 3, 0, 3, 0], [0, 0, 2, 2, 0, 1, 0, 8, 0, 6], [0, 1, 0, 0, 4, 0, 3, 0, 8, 0], [0, 0, 3, 0, 0, 0, 0, 6, 0, 9]], 'component_group': '8.3', 'component_group_number': 3, 'components': 8, 'counts': [['a_1', [[0, 5]]]], 'degree': 4, 'first_a2_moment': 1, 'fourth_trace_moment': 6, 'gens': [[['\\zeta_8', '0', '0', '0'], ['0', '\\zeta_8', '0', '0'], ['0', '0', '\\zeta_8^7', '0'], ['0', '0', '0', '\\zeta_8^7']], [['0', '0', '0', '1'], ['0', '0', '-1', '0'], ['0', '-1', '0', '0'], ['1', '0', '0', '0']]], 'identity_component': 'SU(2)_2', 'label': '1.4.E.8.3a', 'label_components': [1, 4, 4, 8, 3, 0], 'maximal': True, 'moments': [['a_1', 1, 0, 1, 0, 6, 0, 50, 0, 504, 0, 5712, 0, 69696], ['a_2', 1, 1, 3, 7, 26, 96, 422, 1926, 9326, 46402, 236810, 1229394, 6474228]], 'name': 'J(E_4)', 'old_label': '1.4.3.8.3a', 'pretty': 'J(E_4)', 'rational': True, 'real_dimension': 3, 'second_trace_moment': 1, 'simplex': [1, 1, 3, 3, 6, 7, 11, 23, 50, 26, 45, 98, 220, 504, 96, 198, 446, 1028, 2408, 5712, 422, 918, 2124, 5000, 11920, 28704, 69696], 'st0_label': '1.4.E', 'subgroup_multiplicities': [1, 2], 'subgroups': ['1.4.E.4.1a', '1.4.E.4.2a'], 'supgroups': [], 'trace_histogram': 'data:image/png;base64,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', 'trace_zero_density': '5/8', 'weight': 1, 'zvector': [5, 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}A&0\\\\0&\\bar{A}\\end{bmatrix}: A\\in \\mathrm{SU}(2)\\right\\}', 'hodge_circle': 'u\\mapsto\\mathrm{diag}(u,\\bar u,\\bar u,u)', 'label': '1.4.E', 'label_components': [1, 4, 4], 'name': 'SU(2)_2', 'pretty': '\\mathrm{SU}(2)_2', 'real_dimension': 3, '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_abelian	boolean
  aut_cyclic	boolean
  aut_derived_length	smallint
  aut_exponent	numeric
  aut_gen_orders	numeric[]
  aut_gens	numeric[]
  aut_group	text
  aut_hash	bigint
  aut_nilpotency_class	smallint
  aut_nilpotent	boolean
  aut_order	numeric
  aut_permdeg	integer
  aut_perms	numeric[]
  aut_phi_ratio	double precision
  aut_solvable	boolean
  aut_stats	numeric[]
  aut_supersolvable	boolean
  aut_tex	text
  autcent_abelian	boolean
  autcent_cyclic	boolean
  autcent_exponent	numeric
  autcent_group	text
  autcent_hash	bigint
  autcent_nilpotent	boolean
  autcent_order	numeric
  autcent_solvable	boolean
  autcent_split	boolean
  autcent_supersolvable	boolean
  autcent_tex	text
  autcentquo_abelian	boolean
  autcentquo_cyclic	boolean
  autcentquo_exponent	numeric
  autcentquo_group	text
  autcentquo_hash	bigint
  autcentquo_nilpotent	boolean
  autcentquo_order	numeric
  autcentquo_solvable	boolean
  autcentquo_supersolvable	boolean
  autcentquo_tex	text
  cc_stats	numeric[]
  center_label	text
  center_order	numeric
  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
  conjugacy_classes_known	boolean
  counter	integer
  cyclic	boolean
  derived_length	smallint
  dihedral	boolean
  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[]
  familial	boolean
  frattini_label	text
  frattini_quotient	text
  hash	bigint
  hyperelementary	integer
  inner_abelian	boolean
  inner_cyclic	boolean
  inner_exponent	numeric
  inner_gen_orders	numeric[]
  inner_gens	numeric[]
  inner_hash	bigint
  inner_nilpotent	boolean
  inner_order	numeric
  inner_split	boolean
  inner_tex	text
  inner_used	smallint[]
  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_abelian	boolean
  outer_cyclic	boolean
  outer_equivalence	boolean
  outer_exponent	numeric
  outer_gen_orders	numeric[]
  outer_gen_pows	numeric[]
  outer_gens	numeric[]
  outer_group	text
  outer_hash	bigint
  outer_nilpotent	boolean
  outer_order	numeric
  outer_permdeg	integer
  outer_perms	numeric[]
  outer_solvable	boolean
  outer_supersolvable	boolean
  outer_tex	text
  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': False, 'Zgroup': False, 'abelian': False, 'abelian_quotient': '4.2', 'all_subgroups_known': True, 'almost_simple': False, 'aut_abelian': False, 'aut_cyclic': False, 'aut_derived_length': 2, 'aut_exponent': 4, 'aut_gen_orders': [2, 4], 'aut_gens': [[1, 2], [1, 6], [3, 2]], 'aut_group': '8.3', 'aut_hash': 3, 'aut_nilpotency_class': 2, 'aut_nilpotent': True, 'aut_order': 8, 'aut_permdeg': 4, 'aut_perms': [5, 9], 'aut_phi_ratio': 2.0, 'aut_solvable': True, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 2, 1], [4, 2, 1, 1]], 'aut_supersolvable': True, 'aut_tex': 'D_4', 'autcent_abelian': True, 'autcent_cyclic': False, 'autcent_exponent': 2, 'autcent_group': '4.2', 'autcent_hash': 2, 'autcent_nilpotent': True, 'autcent_order': 4, 'autcent_solvable': True, 'autcent_split': True, 'autcent_supersolvable': True, 'autcent_tex': 'C_2^2', 'autcentquo_abelian': True, 'autcentquo_cyclic': True, 'autcentquo_exponent': 2, 'autcentquo_group': '2.1', 'autcentquo_hash': 1, 'autcentquo_nilpotent': True, 'autcentquo_order': 2, 'autcentquo_solvable': True, 'autcentquo_supersolvable': True, 'autcentquo_tex': 'C_2', 'cc_stats': [[1, 1, 1], [2, 1, 1], [2, 2, 2], [4, 2, 1]], 'center_label': '2.1', 'center_order': 2, 'central_product': False, 'central_quotient': '4.2', 'commutator_count': 1, 'commutator_label': '2.1', 'complements_known': True, 'complete': False, 'complex_characters_known': True, 'composition_factors': ['2.1', '2.1', '2.1'], 'composition_length': 3, 'conjugacy_classes_known': True, 'counter': 3, 'cyclic': False, 'derived_length': 2, 'dihedral': True, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [2, 2, 1, 2], [4, 2, 1, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 3, 'exponent': 4, 'exponents_of_order': [3], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[2, 1, 1]], 'familial': True, 'frattini_label': '2.1', 'frattini_quotient': '4.2', 'hash': 3, 'hyperelementary': 2, 'inner_abelian': True, 'inner_cyclic': False, 'inner_exponent': 2, 'inner_gen_orders': [2, 2], 'inner_gens': [[1, 6], [5, 2]], 'inner_hash': 2, 'inner_nilpotent': True, 'inner_order': 4, 'inner_split': True, 'inner_tex': 'C_2^2', 'inner_used': [1, 2], 'irrC_degree': 2, 'irrQ_degree': 2, 'irrQ_dim': 2, 'irrR_degree': 2, 'irrep_stats': [[1, 4], [2, 1]], 'label': '8.3', 'linC_count': 1, 'linC_degree': 2, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'linQ_degree_count': 1, 'linQ_dim': 2, 'linQ_dim_count': 1, 'linR_count': 1, 'linR_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'D4', 'ngens': 2, 'nilpotency_class': 2, 'nilpotent': True, 'normal_counts': [0, 0, 0, 0], 'normal_index_bound': 0, 'normal_order_bound': 0, 'normal_subgroups_known': True, 'number_autjugacy_classes': 4, 'number_characteristic_subgroups': 4, 'number_conjugacy_classes': 5, 'number_divisions': 5, 'number_normal_subgroups': 6, 'number_subgroup_autclasses': 6, 'number_subgroup_classes': 8, 'number_subgroups': 10, 'old_label': None, 'order': 8, 'order_factorization_type': 3, 'order_stats': [[1, 1], [2, 5], [4, 2]], 'outer_abelian': True, 'outer_cyclic': True, 'outer_equivalence': False, 'outer_exponent': 2, 'outer_gen_orders': [2], 'outer_gen_pows': [2], 'outer_gens': [[3, 2]], 'outer_group': '2.1', 'outer_hash': 1, 'outer_nilpotent': True, 'outer_order': 2, 'outer_permdeg': 2, 'outer_perms': [1], 'outer_solvable': True, 'outer_supersolvable': True, 'outer_tex': 'C_2', 'pc_rank': 2, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [2, 2], 'quasisimple': False, 'rank': 2, 'rational': True, 'rational_characters_known': True, 'ratrep_stats': [[1, 4], [2, 1]], 'representations': {'PC': {'code': 294, 'gens': [1, 2], 'pres': [3, -2, 2, -2, 37, 16]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [14, 46]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [29, 56, 24], 'family': 'COPlus'}, {'d': 1, 'q': 4, 'gens': [7, 16, 1], 'family': 'ASigmaL'}], 'GLFp': {'d': 2, 'p': 3, 'gens': [12, 55, 56]}, 'Perm': {'d': 4, 'gens': [6, 16, 7]}}, 'schur_multiplier': [2], 'semidirect_product': True, 'simple': False, 'smith_abelian_invariants': [2, 2], 'solvability_type': 3, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'D_4', 'transitive_degree': 4, 'wreath_data': ['C_2', 'C_2', '2T1'], 'wreath_product': True}