{"rank":3,"n":11,"seq":208972,"label":"3.11.208972","id":"r3_n11_0000************0*********0***0***0*******0****0***************0**************0**0**********0****0**************0******************0*******************0***********0*","beta_invariant":20,"birational_type":"rational over Q(a)/(a^3 - a^2 + 1)","birational_type_components":"[{\"dim\":1,\"free_rank\":1,\"torus_rank\":1,\"core_dim\":0,\"qbar_components\":3,\"type\":\"rational over Q(a)/(a^3 - a^2 + 1)\",\"field\":\"Q(a)/(a^3 - a^2 + 1)\",\"minpoly\":\"T^3 - T^2 + 1\",\"degree\":3,\"reason\":\"A^1 over the field of definition\",\"core_vars\":[\"x2\"],\"core_gens\":[\"x2^3 - x2^2 + 1\"]}]","birational_type_method":"point_field","char_poly_splits":false,"char_set":"[0]","expected_dimension_over_Z":2,"free_realization_space":false,"girth":3,"good_basis":"1,2,5","good_basis_v2":"1,2,5","is_binary":false,"is_coloopless":true,"is_connected":true,"is_divisionally_free":false,"is_graphic":false,"is_identically_self_dual":false,"is_laminar":false,"is_loopless":true,"is_nested":false,"is_orientable":true,"is_paving":true,"is_quaternary":false,"is_realizable":true,"is_realizable_char0":true,"is_regular":false,"is_self_dual":false,"is_series_parallel":false,"is_simple":true,"is_simplicial_arrangement":false,"is_supersolvable":false,"is_ternary":false,"is_three_lines":false,"is_transversal":false,"is_uniform":null,"n_automorphisms":1,"n_bases":146,"n_circuits":206,"n_components":null,"n_connected_components":1,"n_cyclic_flats":12,"n_flats":39,"n_loops":0,"principal_ideal":true,"rs_characteristic_dimension_unexpected":false,"rs_characteristic_dimension_varies":false,"rs_characteristic_dimensions":"[{\"p\":0,\"d\":1}]","rs_characteristic_dimensions_method":"cofinite_strong_groebner_candidates_exact","rs_dim":2,"rs_is_regular_scheme":true,"rs_is_smooth_over_QQ":null,"rs_is_smooth_over_ZZ":false,"rs_n_qbar_components":null,"rs_regularity_method":"three_lines_deletion","rs_regularity_witness":"7=>r3_n10_0000************0*********0***0***0*******0****0****************0****0*************0****************0*******0***0*******","rs_scheme_simple_core_id":"r3_n11_0000************0*********0***0***0*******0****0***************0**************0**0**********0****0**************0******************0*******************0***********0*","rs_singular_fiber_characteristics":null,"rs_singular_fiber_characteristics_method":null,"rs_singular_fiber_primes":"[23]","rs_singular_fiber_primes_method":"smooth_deletion_sandwich","rs_smoothness_method":"certified_singular_fiber","rs_smoothness_witness":"23","singular_primes":"[23]","smooth_char_0":true,"status":"computed_realization_space","tutte_polynomial":"x_0^3 + 8*x_0^2 + 3*x_0*x_1^2 + 13*x_0*x_1 + 20*x_0 + x_1^8 + 3*x_1^7 + 6*x_1^6 + 10*x_1^5 + 15*x_1^4 + 21*x_1^3 + 25*x_1^2 + 20*x_1","realization_space":{"vars":["x1","x2"],"ideal":["x2^3 - x2^2 + 1"],"inequations":["x1 - x2 - 1","x1 + x2^2 - 1","x1","x1 - 1","x1*x2 - x1 + 1","x1*x2 - x1 - x2","x1 - x2","x1*x2^2 - x1 - x2^3","x1 + x2","x1 + x2^3","x1 - x2^2"],"matrix":[["1","0","1","1","0","1","1","0","0","1","1"],["0","1","1","x2^2","0","0","0","1","1","1","x2"],["0","0","0","0","1","1","x1","-1","x2","x2","1"]]}}