-
gps_st • Show schema
{'character_diagonal': [1, 3, 9, 9, 17, 104, 114, 162, 283, 119, 68, 610, 742, 1727, 3424, 1697, 1720, 4146, 3512, 1003], 'character_matrix': [[1, 0, 1, 0, 1, 0, 2, 4, 0, 3, 0, 6, 0, 0, 12], [0, 3, 0, 3, 0, 14, 0, 0, 21, 0, 11, 0, 29, 49, 0], [1, 0, 9, 0, 10, 0, 29, 33, 0, 24, 0, 63, 0, 0, 140], [0, 3, 0, 9, 0, 26, 0, 0, 39, 0, 17, 0, 69, 95, 0], [1, 0, 10, 0, 17, 0, 36, 47, 0, 39, 0, 88, 0, 0, 196], [0, 14, 0, 26, 0, 104, 0, 0, 166, 0, 80, 0, 264, 404, 0], [2, 0, 29, 0, 36, 0, 114, 127, 0, 100, 0, 254, 0, 0, 588], [4, 0, 33, 0, 47, 0, 127, 162, 0, 133, 0, 305, 0, 0, 716], [0, 21, 0, 39, 0, 166, 0, 0, 283, 0, 131, 0, 441, 689, 0], [3, 0, 24, 0, 39, 0, 100, 133, 0, 119, 0, 250, 0, 0, 604], [0, 11, 0, 17, 0, 80, 0, 0, 131, 0, 68, 0, 204, 325, 0], [6, 0, 63, 0, 88, 0, 254, 305, 0, 250, 0, 610, 0, 0, 1420], [0, 29, 0, 69, 0, 264, 0, 0, 441, 0, 204, 0, 742, 1107, 0], [0, 49, 0, 95, 0, 404, 0, 0, 689, 0, 325, 0, 1107, 1727, 0], [12, 0, 140, 0, 196, 0, 588, 716, 0, 604, 0, 1420, 0, 0, 3424]], 'component_group': '4.1', 'component_group_number': 1, 'components': 4, 'counts': [['a_1', [[0, 2]]], ['a_2', [[1, 2]]], ['a_3', [[0, 2]]], ['(a_1,a_2)', [['(0,1)', 2]]], ['(a_1,a_3)', [['(0,0)', 2]]], ['(a_2,a_3)', [['(1,0)', 2]]], ['(a_1,a_2,a_3)', [['(0,1,0)', 2]]]], 'degree': 6, 'first_a2_moment': 2, 'fourth_trace_moment': 51, 'gens': '\\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 1 & 0 & 0 & 0 & 0 \\\\0 & 0 & i & 0 &0 & 0 \\\\0 & 0 & 0 & -i & 0 & 0 \\\\0 & 0 & 0 & 0 & -i & 0 \\\\0 & 0 & 0 & 0 & 0 & i \\\\\\end{bmatrix}, \\begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\\\-1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & \\zeta_{8}^{1} \\\\0 & 0 & 0 & 0 & \\zeta_{8}^{3} & 0 \\\\0 & 0 & 0 & \\zeta_{8}^{3} & 0 & 0 \\\\0 & 0 & \\zeta_{8}^{1} & 0 & 0 & 0 \\\\\\end{bmatrix}', 'identity_component': 'U(1)xU(1)_2', 'label': '1.6.L.4.1a', 'label_components': [1, 6, 11, 4, 1, 0], 'maximal': False, 'moments': [['a_1', 1, 0, 3, 0, 51, 0, 1230, 0, 33635, 0, 978138, 0, 29546286], ['a_2', 1, 2, 12, 113, 1270, 15337, 192897, 2492380, 32839506, 439210589, 5944149907, 81223529290, 1118751586795], ['a_3', 1, 0, 18, 0, 3570, 0, 966600, 0, 292513298, 0, 94093174728, 0, 31472712201480]], 'name': 'L(C_{4,1},C_2)', 'pretty': 'L(C_{4,1},C_2)', 'rational': True, 'real_dimension': 6, 'second_trace_moment': 3, 'simplex': [2, 3, 12, 6, 23, 51, 18, 113, 64, 245, 138, 545, 1230, 182, 1270, 714, 406, 2851, 1604, 6461, 3620, 14710, 33635, 2120, 15337, 1188, 8562, 4790, 34985, 19502, 10900, 80139, 44600, 184102, 102270, 423913, 978138, 3570, 25976, 192897, 14482, 106994, 59440, 444073, 33060, 245982, 136436, 1024753, 566872, 314014, 2369222, 1309000, 5486565, 3027780, 12723858, 29546286], 'st0_label': '1.6.L', 'subgroup_multiplicities': [1], 'subgroups': ['1.6.L.2.1d'], 'supgroup_multiplicities': [1, 1, 1, 1, 1], 'supgroups': ['1.6.L.8.2c', '1.6.L.8.3c', '1.6.L.8.3b', '1.6.L.8.2b', '1.6.L.8.2d'], 'trace_histogram': 'data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAABLAAAAK8AQMAAADF/dTNAAAABGdBTUEAALGPC/xhBQAAACBjSFJNAAB6JgAAgIQAAPoAAACA6AAAdTAAAOpgAAA6mAAAF3CculE8AAAABlBMVEX///8AcrJtpBqqAAAAAWJLR0QAiAUdSAAAAAd0SU1FB+UBGA0GE/isA1oAAAaISURBVHja7dxviBR1HMfxmd1rd+U4bs9QLqTuhHtgSAhlpcR1G/7hREzB/kEPTokoIggy8oHRjtITwVKjohLDBxVBgRaIpl27gRiCpoEpeupuaRmY7fXHO3Xd/bZ7pd62M3e7s/Pnt3vvz6PdmV2+L2bm95vfzP52NK3qGBbLdcsV1dewEViwYMGCBQsWLFiwYMGCBQsWLFiqsZKw3GQFYcGCBQsWLFiwYMGCBQsWLFiwYMGCBQsWLFiwYMGCBQsWLFiwYMGCBQsWLFiwYMGCBQsWLFiwYMGCBQsWLFiwYMGCBQsWLFiwYMGCBQsWLFiwYMGCBQsWLFiwYMGCBQsWLFiwYMGCBQsWLFiwYMGCBQsWLFiwYMGCBQsWLFiwYMGCBcuhpGHBggXLMVYYFixYsGDBggULFixYsGDBggULFixYsGDBggULFixYsGDBggULFixYsGDBggWrUViDsGDBgtXorFZYsGDBggULFixYsGDBggULFixYsGDBggULFqwJzLoMC5YqrA5YsGDBggULFixY1eQaLFiwYDnG6oEFCxYsWLBgwYIFCxYsWLBgwWoYVr5BWHFYsGDBggULFixFWQJrArKsvgALFixYsGDBgmXJ0mEpyLIqDwsWLC9ZSVjusYKwYMGCZcUKT2xWGhYsWLBgwbqeVo9YFv/JhQUL1kg6YFXB6oEFCxYsWPXEuqomy2IWHqwGZsVhwXKJJRObZVEGVl2xzOtbsIKw6p8V9oyVhAVrYrHS1bBaPWMNwoIFy5I1S01W3DOW+Z0kWOYs8+dnwILVsCzzmxDmrKCiLPGMZV7Hf5YpwH9WElb9s9Kw3GGF1WS1esj6VU2W6fDUnNUBqwpWn4esK2qyspWz4h6yTAvVFUtgmQ4h6okVhCVyTE3WmUpZ93jKMrtSNGOFHlCT1eMpy6wSLEtWrEJWwlvWirpmhcVblsn4FFatrFmw6p+13WOWyQRiM1bGY5Zsq4R1i3jNSlfCCqvJ6oElFk2xnBXynlXeFJVg5csQJizxnFX+A0s5K+wD6zU1Wf1acjzWMz6wck3jss77wZo/DmtqUHxgybis+31hDanJyo/H2uUTKzo2K+ELS/aNzbpD/GENtY/B0kduffvByt+mJEveG4vV4hsrVQopfXeX+MUqHUWUsEL6C/6xDlqzNsb9Y5XMmPofS/xj5aOWrIyfrEJbbCpnFV7l/WRJalHMnDX6Q96zZHHvI2WsmL7hOZ9ZFw4e7TRhXfCZlT10NF3OOpXxmSVy7OymUaziaXL/zNPiO0sym7XOm6yArn2bFQVY0t/+L2utphm3vzj3y4wowZJp25ZoWrLA6jS6s3MToghLdi3dMSc5b/3KHUa3dIsyrMTS/t3fne3PfPFb4fpRHZakctlMATT8u9lK/1g3WqWaLIEFCxYs9Vg6LFiwYMHyiWXAggULFqx6YQUbhpWcuKxw9ax9arLOqcn6W03WtdqrjpulDcMqn7DnfFIqsiKB6lUedPOTWu2wkmqy0mqyXO9PWzrssK51uswaPRmk8uRc7rn0w3ZUrneoesoeK/+Qq6xJNlmy0oi4tqkMvcWmSlaud4016ZMDNg8tkeyVd11SLWzuv2CbJdmTmuECKhra35GwjSpmwefRmPOsqYfjtbEO7RxhdTqp0pd/kKoJVUhiR4FlOMp6/XiiZpbsOnLCMVZU17SAtvV4zSiR4ZMnpr3vDMto07vmN99Z+6YqJpWSgemaEaidtW7zgfMPr3EEVXSJ7NxjhAIxQ9NHTSusOCtihd0XMZ7XftqSO1+zZnQSe978eHJssMkuqy06x3hldmYgVzulNHlZ3nspFAws66rOFIh0Pp1uapv84IbcS5Kt3VHGWnVVeraHHuvSm9tiFZ4rDa0pMKPr4o8dfa/mU+L4prqxL2Uo/9mi5rcfjURC0VHloxastZEiS5w9oMxYwyJrFm/9YUb7G7MLB3Kh9x8Z+5SyAsXldzdpXe3f7H55/957XWeNJBvPyZUtG7ufajK0FZEia3IJKxYIaV299y1oGXri8GX5JbvaC1Qh8eJBkkqsfuv7Kc/OMHq12E1WdLoWXTZz4boPT8WPL/lvClbcI9b1nBk4ffGrn/vW/nHrEW1esrj3ou989OSU/Kf5vfmcOHGGsZeBXCZ3Sf7Krfpz5uPnNkW1YDyRkL5Cq3W+I7CXNcOZwdkjTxNTLbmvbcyY8SKwYMFSIbBgwVIhsGDBUiGwGoD1D6ZmcUEnAhimAAAAJXRFWHRkYXRlOmNyZWF0ZQAyMDIxLTAxLTI0VDE4OjA2OjE5LTA1OjAwvvxKFAAAACV0RVh0ZGF0ZTptb2RpZnkAMjAyMS0wMS0yNFQxODowNjoxOS0wNTowMM+h8qgAAAAASUVORK5CYII=', 'trace_zero_density': '1/2', 'weight': 1, 'zvector': [2, 0, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0]} -
gps_st0 • Show schema
{'degree': 6, 'description': '\\left\\{\\begin{bmatrix}A&0&0\\\\0&\\alpha I_2&0\\\\0&0&\\bar\\alpha I_2\\end{bmatrix}: A\\in\\mathrm{U}(1)\\subseteq\\mathrm{SU}(2),\\ \\alpha\\bar\\alpha = 1,\\ \\alpha\\in\\mathbb{C}\\right\\}', 'hodge_circle': 'u\\mapsto\\mathrm{diag}(u,\\bar u, u, u,\\bar u, \\bar u)', 'label': '1.6.L', 'label_components': [1, 6, 11], 'name': 'U(1)xU(1)_2', 'pretty': '\\mathrm{U}(1)\\times\\mathrm{U}(1)_2', 'real_dimension': 2, 'symplectic_form': '\\begin{bmatrix}J_2&0&0\\\\0&0&I_2\\\\0&-I_2&0\\end{bmatrix},\\ J_2:=\\begin{bmatrix}0&1\\\\-1&0\\end{bmatrix}', 'weight': 1} -
gps_groups • Show schema
{'Agroup': True, 'Zgroup': True, 'abelian': True, 'abelian_quotient': '4.1', 'all_subgroups_known': True, 'almost_simple': False, 'aut_gens': [[1], [3]], 'aut_group': '2.1', 'aut_order': 2, 'aut_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1]], 'cc_stats': [[1, 1, 1], [2, 1, 1], [4, 1, 2]], 'center_label': '4.1', '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', '2.1'], 'composition_length': 2, 'counter': 1, 'cyclic': True, 'derived_length': 1, 'direct_factorization': [], 'direct_product': False, 'div_stats': [[1, 1, 1, 1], [2, 1, 1, 1], [4, 1, 2, 1]], 'element_repr_type': 'PC', 'elementary': 2, 'eulerian_function': 1, 'exponent': 4, 'exponents_of_order': [2], 'factors_of_aut_order': [2], 'factors_of_order': [2], 'faithful_reps': [[1, 0, 2]], 'frattini_label': '2.1', 'frattini_quotient': '2.1', 'hash': 1, 'hyperelementary': 2, 'irrC_degree': 1, 'irrQ_degree': 2, 'irrep_stats': [[1, 4]], 'label': '4.1', 'linC_degree': 1, 'linFp_degree': None, 'linFq_degree': None, 'linQ_degree': 2, 'maximal_subgroups_known': True, 'metabelian': True, 'metacyclic': True, 'monomial': True, 'name': 'C4', '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': 3, 'number_characteristic_subgroups': 3, 'number_conjugacy_classes': 4, 'number_divisions': 3, 'number_normal_subgroups': 3, 'number_subgroup_autclasses': 3, 'number_subgroup_classes': 3, 'number_subgroups': 3, 'old_label': None, 'order': 4, 'order_factorization_type': 2, 'order_stats': [[1, 1], [2, 1], [4, 2]], 'outer_equivalence': False, 'outer_group': '2.1', 'outer_order': 2, 'pc_rank': 1, 'perfect': False, 'permutation_degree': 4, 'pgroup': 2, 'primary_abelian_invariants': [4], 'quasisimple': False, 'rank': 1, 'rational': False, 'rational_characters_known': True, 'ratrep_stats': [[1, 2], [2, 1]], 'representations': {'PC': {'code': 5, 'gens': [1], 'pres': [2, -2, -2, 4]}, 'GLZ': {'b': 3, 'd': 2, 'gens': [46]}, 'Lie': [{'d': 2, 'q': 3, 'gens': [56, 15], 'family': 'CSOPlus'}], 'GLFp': {'d': 2, 'p': 3, 'gens': [21]}, 'Perm': {'d': 4, 'gens': [22, 7]}}, 'schur_multiplier': [], 'semidirect_product': False, 'simple': False, 'smith_abelian_invariants': [4], 'solvability_type': 0, 'solvable': True, 'subgroup_inclusions_known': True, 'subgroup_index_bound': 0, 'supersolvable': True, 'sylow_subgroups_known': True, 'tex_name': 'C_4', 'transitive_degree': 4, 'wreath_data': None, 'wreath_product': False}
Sato-Tate group data - 1.6.L.4.1a