{"rank":3,"n":11,"seq":97648,"label":"3.11.97648","id":"r3_n11_0000************0**********0***************0***********0*********0*****************************0***0*******************0*************0********0*******0*********0****","beta_invariant":21,"birational_type":"point over Q(a)/(a^3 - 2*a - 2)","birational_type_components":"[{\"dim\":0,\"free_rank\":0,\"torus_rank\":0,\"core_dim\":0,\"qbar_components\":3,\"type\":\"point over Q(a)/(a^3 - 2*a - 2)\",\"field\":\"Q(a)/(a^3 - 2*a - 2)\",\"minpoly\":\"T^3 - 2*T - 2\",\"degree\":3,\"core_vars\":[\"x1\"],\"core_gens\":[\"3*x1^3 - 8*x1^2 + 8*x1 - 4\"]}]","birational_type_method":"point_field","char_poly_splits":false,"char_set":"[0]","expected_dimension_over_Z":1,"free_realization_space":false,"girth":3,"good_basis":"1,2,5","good_basis_v2":"2,4,5","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":true,"is_transversal":false,"is_uniform":null,"n_automorphisms":1,"n_bases":149,"n_circuits":221,"n_components":null,"n_connected_components":1,"n_cyclic_flats":15,"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\":0}]","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":"direct_mixed_jacobian","rs_regularity_witness":null,"rs_scheme_simple_core_id":"r3_n11_0000************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":"[19]","rs_singular_fiber_primes_method":"terminal_three_lines_integer_certificate","rs_smoothness_method":"certified_singular_fiber","rs_smoothness_witness":"19","singular_primes":"[19]","smooth_char_0":true,"status":"computed_realization_space","tutte_polynomial":"x_0^3 + 8*x_0^2 + x_0*x_1^2 + 14*x_0*x_1 + 21*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 + 27*x_1^2 + 21*x_1","realization_space":{"vars":["x1"],"ideal":["3*x1^3 - 8*x1^2 + 8*x1 - 4"],"inequations":["3*x1 - 2","3*x1^2 - 5*x1 + 4","3*x1^2 - 8*x1 + 8","3*x1^3 - 5*x1^2 + 2*x1 - 2","3*x1^2 - 2*x1 + 2","x1","x1 - 2","3*x1^2 - 2*x1 - 2","9*x1^4 - 30*x1^3 + 34*x1^2 - 18*x1 + 8","27*x1^4 - 90*x1^3 + 126*x1^2 - 88*x1 + 32","9*x1^3 - 21*x1^2 + 22*x1 - 14","9*x1^4 - 30*x1^3 + 40*x1^2 - 28*x1 + 16","2","3*x1^2 - 2*x1 + 4","3*x1^2 - 2*x1 - 4","3*x1^3 - 8*x1^2 + 7*x1 - 4","3*x1^3 - 14*x1^2 + 18*x1 - 8","3*x1^3 - 11*x1^2 + 11*x1 - 6","9*x1^4 - 39*x1^3 + 64*x1^2 - 52*x1 + 24","18*x1^4 - 69*x1^3 + 110*x1^2 - 84*x1 + 32","9*x1^3 - 24*x1^2 + 24*x1 - 16","6*x1^2 - 7*x1 + 6","x1^2 - 2*x1 + 2","3*x1^2 - 8*x1 + 6","3*x1^3 - 14*x1^2 + 14*x1 - 8","3*x1^2 - 4*x1 + 2","3*x1^3 - 8*x1^2 + 4*x1 - 4","3*x1^2 - 4*x1 + 4","3*x1^3 - 2*x1^2 - 4*x1 + 4","9*x1^4 - 12*x1^3 - 14*x1^2 + 24*x1 - 16","6*x1^2 - 7*x1 + 4","9*x1^4 - 12*x1^3 + 4*x1^2 - 8"],"matrix":[["1","1","4","0","0","1","2","4","0","2","2"],["1","0","3*x1^2 - 2*x1","1","0","1","0","3*x1^2 - 2*x1","1","3*x1^2 - 5*x1 + 4","3*x1^2 - 5*x1 + 4"],["0","0","0","0","1","1","-3*x1^2 + 2*x1 - 2","6*x1^2 - 10*x1 + 4","x1","3*x1^2 - 5*x1 + 4","2"]]}}