{"rank":3,"n":11,"seq":46844,"label":"3.11.46844","id":"r3_n11_0******0******0***0******0****************0****************************0*******0********************0*******0********0****************************0**0******0********","beta_invariant":22,"birational_type":"point over Q(a)/(a^4 - a - 1)","birational_type_components":"[{\"dim\":0,\"free_rank\":0,\"torus_rank\":0,\"core_dim\":0,\"qbar_components\":4,\"type\":\"point over Q(a)/(a^4 - a - 1)\",\"field\":\"Q(a)/(a^4 - a - 1)\",\"minpoly\":\"T^4 - T - 1\",\"degree\":4,\"core_vars\":[\"x1\"],\"core_gens\":[\"x1^4 - 4*x1^3 + 6*x1^2 - 3*x1 - 1\"]}]","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":"1,2,4","good_basis_v2":"1,2,4","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":151,"n_circuits":232,"n_components":null,"n_connected_components":1,"n_cyclic_flats":16,"n_flats":40,"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":0,"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_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":"[283]","rs_singular_fiber_primes_method":"terminal_three_lines_integer_certificate","rs_smoothness_method":"certified_singular_fiber","rs_smoothness_witness":"283","singular_primes":"[283]","smooth_char_0":true,"status":"computed_realization_space","tutte_polynomial":"x_0^3 + 8*x_0^2 + 14*x_0*x_1 + 22*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 + 28*x_1^2 + 22*x_1","realization_space":{"vars":["x1"],"ideal":["x1^4 - 4*x1^3 + 6*x1^2 - 3*x1 - 1"],"inequations":["x1^2 - x1 + 1"],"matrix":[["1","0","1","0","1","0","x1^2 - 2*x1 + 1","x1 - 1","x1^2 - 2*x1 + 1","x1^2 - 2*x1 + 1","x1 - 1"],["0","1","1","0","0","1","x1^2 - 2*x1 + 1","1","x1^3 - 3*x1^2 + 2*x1 + 1","x1 - 1","x1^2 - x1"],["0","0","0","1","1","-1","-x1 + 2","x1 - 1","-x1^3 + 3*x1^2 - 2*x1","-x1 + 2","-x1^2 + 2*x1"]]}}