{"rank":3,"n":10,"seq":6712,"label":"3.10.6712","id":"r3_n10_0000000000********************0***************0*********************0**************0****************0******0****0*******","beta_invariant":15,"birational_type":"rational over Q(a)/(a^2 - a + 1)","birational_type_components":"[{\"dim\":2,\"free_rank\":2,\"torus_rank\":2,\"core_dim\":0,\"qbar_components\":2,\"type\":\"rational over Q(a)/(a^2 - a + 1)\",\"field\":\"Q(a)/(a^2 - a + 1)\",\"minpoly\":\"T^2 - T + 1\",\"degree\":2,\"reason\":\"A^2 over the field of definition\",\"core_vars\":[\"x3\"],\"core_gens\":[\"x3^2 - x3 + 1\"]}]","birational_type_method":"point_field","char_poly_splits":false,"char_set":"[0]","expected_dimension_over_Z":3,"free_realization_space":false,"girth":3,"good_basis":"1,2,6","good_basis_v2":"1,2,8","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":false,"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":12,"n_bases":103,"n_circuits":123,"n_components":2,"n_connected_components":1,"n_cyclic_flats":10,"n_flats":34,"n_loops":0,"principal_ideal":true,"rs_characteristic_dimension_unexpected":false,"rs_characteristic_dimension_varies":false,"rs_characteristic_dimensions":"[{\"p\":0,\"d\":2}]","rs_characteristic_dimensions_method":"cofinite_strong_groebner_candidates_exact","rs_dim":5,"rs_is_regular_scheme":true,"rs_is_smooth_over_QQ":null,"rs_is_smooth_over_ZZ":false,"rs_n_qbar_components":2,"rs_regularity_method":"three_lines_deletion","rs_regularity_witness":"4=>r3_n9_0000************0**********0***************0***********0***********0*****0***0******","rs_scheme_simple_core_id":"r3_n10_0000000000********************0***************0*********************0**************0****************0******0****0*******","rs_singular_fiber_characteristics":null,"rs_singular_fiber_characteristics_method":null,"rs_singular_fiber_primes":"[3]","rs_singular_fiber_primes_method":"smooth_deletion_sandwich","rs_smoothness_method":"certified_singular_fiber","rs_smoothness_witness":"3","singular_primes":"[3]","smooth_char_0":true,"status":"computed_realization_space","tutte_polynomial":"x_0^3 + 7*x_0^2 + x_0*x_1^3 + 2*x_0*x_1^2 + 10*x_0*x_1 + 15*x_0 + x_1^7 + 3*x_1^6 + 6*x_1^5 + 10*x_1^4 + 14*x_1^3 + 18*x_1^2 + 15*x_1","realization_space":{"vars":["x1","x2","x3"],"ideal":["x3^2 - x3 + 1"],"inequations":["x1 - x3^2 + x3 - 1","x2 - x3^2 + x3 - 1","x1 - 1","x2 - 1","x1 + x3 - 1","x2 + x3 - 1","x2","x1 - x2","x1","x1 - x3","x2 - x3"],"matrix":[["1","0","1","1","1","0","1","0","1","1"],["0","1","1","x1","x2","1","x3","0","x3","1"],["0","0","0","0","0","1","x3","1","x3 - 1","x3"]]}}