{"rank":3,"n":11,"seq":26546,"label":"3.11.26546","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^3 + 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^3 + a + 1)\",\"field\":\"Q(a)/(a^4 - a^3 + a + 1)\",\"minpoly\":\"T^4 - T^3 + T + 1\",\"degree\":4,\"core_vars\":[\"x1\"],\"core_gens\":[\"x1^4 + 3*x1^3 + x1^2 - 8*x1 + 4\"]}]","birational_type_method":"point_field","char_poly_splits":false,"char_set":"[0,2]","expected_dimension_over_Z":1,"free_realization_space":false,"girth":3,"good_basis":"2,3,4","good_basis_v2":"2,8,10","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":true,"is_transversal":false,"is_uniform":null,"n_automorphisms":2,"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":"[7]","rs_singular_fiber_primes_method":"terminal_three_lines_integer_certificate","rs_smoothness_method":"certified_singular_fiber","rs_smoothness_witness":"7","singular_primes":"[7]","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 + 3*x1^3 + x1^2 - 8*x1 + 4"],"inequations":["x1^2 + 2*x1 - 2","x1^2 + 4*x1 - 4","2","x1","x1^4 + 4*x1^3 - 2*x1^2 - 6*x1 + 4","2*x1^3 + 5*x1^2 - 10*x1 + 4","x1 - 2","x1 + 2","2*x1^4 + 6*x1^3 - 7*x1^2 - 4*x1 + 4","2*x1^2 + x1 - 2","2*x1^3 + 3*x1^2 - 8*x1 + 4","x1^5 + 3*x1^4 + x1^3 + 10*x1^2 - 20*x1 + 8","x1^4 + 2*x1^3 + 24*x1^2 - 40*x1 + 16","5*x1^4 + 16*x1^3 - 18*x1^2 - 8*x1 + 8","x1^4 + 12*x1^2 - 20*x1 + 8","3*x1 - 2","2*x1^3 + 2*x1^2 - 5*x1 + 2","x1^3 + 4*x1^2 + 4*x1 - 4","x1^4 + 2*x1^3 - 2*x1^2 - 6*x1 + 4","x1^4 + 3*x1^3 + 10*x1^2 - 20*x1 + 8","2*x1^4 + 6*x1^3 - x1^2 - 14*x1 + 8","2*x1^5 + 5*x1^4 - 12*x1^3 - 20*x1^2 + 40*x1 - 16","2*x1^6 - 21*x1^4 + 2*x1^3 + 56*x1^2 - 56*x1 + 16","x1^5 + 2*x1^4 - 2*x1^3 - 18*x1^2 + 24*x1 - 8","3*x1^5 + 8*x1^4 - 2*x1^3 - 40*x1^2 + 48*x1 - 16","x1^4 - 4*x1^3 - 4*x1^2 + 10*x1 - 4","5*x1^3 + 4*x1^2 - 16*x1 + 8"],"matrix":[["1","1","x1","3*x1^3 - 2*x1^2","3*x1 - 2","x1","0","0","1","0","x1"],["1","0","-x1^2 - x1 + 2","5*x1^3 + 22*x1^2 - 40*x1 + 16","-2*x1^2 - x1 + 2","-x1^2 - x1 + 2","x1","1","0","0","x1"],["1","0","-x1^2 - x1 + 2","-11*x1^3 - 3*x1^2 + 24*x1 - 12","0","-x1 + 2","-x1^2 - x1 + 2","0","x1","1","-x1 + 2"]]}}