{"rank":3,"n":11,"seq":128760,"label":"3.11.128760","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^4 - a^3 + 2*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 + 2*a + 1)\",\"field\":\"Q(a)/(a^4 - a^3 + 2*a + 1)\",\"minpoly\":\"T^4 - T^3 + 2*T + 1\",\"degree\":4,\"core_vars\":[\"x1\"],\"core_gens\":[\"x1^4 - 3*x1^3 + 6*x1^2 - 6*x1 + 3\"]}]","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":"1,4,5","good_basis_v2":"1,4,5","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":1,"rs_is_regular_scheme":true,"rs_is_smooth_over_QQ":null,"rs_is_smooth_over_ZZ":true,"rs_n_qbar_components":null,"rs_regularity_method":"smooth","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":"[]","rs_singular_fiber_primes_method":"smooth_over_ZZ","rs_smoothness_method":"three_lines_exact_bad_prime_certificate","rs_smoothness_witness":null,"singular_primes":"[]","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":["x1^4 - 3*x1^3 + 6*x1^2 - 6*x1 + 3"],"inequations":["x1","x1^2 - 2*x1 + 3","x1 - 1","x1^4 - 3*x1^3 + 3*x1^2 - 3","x1^4 - 3*x1^3 + 9*x1^2 - 12*x1 + 9","2*x1^2 - 3*x1 + 3","x1^4 - 2*x1^3 + 3*x1^2 - 3*x1 + 3","x1^4 - 3*x1^3 + 7*x1^2 - 9*x1 + 6","x1^6 - 3*x1^5 + 7*x1^4 - 12*x1^3 + 15*x1^2 - 12*x1 + 6","x1^3 - 2*x1^2 + 3*x1 - 3","x1 - 2","x1^2 - x1 + 2","x1^4 - 2*x1^3 + 4*x1^2 - 4*x1 + 3","x1^5 - 5*x1^4 + 10*x1^3 - 14*x1^2 + 12*x1 - 6","2*x1^3 - 4*x1^2 + 6*x1 - 3","x1^2 - 3*x1 + 3","2*x1^5 - 6*x1^4 + 13*x1^3 - 16*x1^2 + 12*x1 - 3","x1^4 - 2*x1^3 + 5*x1^2 - 6*x1 + 3","x1^2 - x1 + 3","x1^6 - 4*x1^5 + 12*x1^4 - 23*x1^3 + 31*x1^2 - 27*x1 + 12","x1^3 + 3","x1^5 - 3*x1^4 + 8*x1^3 - 10*x1^2 + 9*x1 - 3","x1^6 - 4*x1^5 + 11*x1^4 - 18*x1^3 + 20*x1^2 - 14*x1 + 6","x1^6 - 4*x1^5 + 10*x1^4 - 16*x1^3 + 17*x1^2 - 12*x1 + 6","x1^6 - 4*x1^5 + 10*x1^4 - 15*x1^3 + 16*x1^2 - 12*x1 + 6","x1^5 - 3*x1^4 + 7*x1^3 - 9*x1^2 + 9*x1 - 6","x1^5 - 3*x1^4 + 8*x1^3 - 11*x1^2 + 12*x1 - 6","x1^5 - 4*x1^4 + 10*x1^3 - 14*x1^2 + 12*x1 - 3","x1^4 - 4*x1^3 + 7*x1^2 - 9*x1 + 3"],"matrix":[["1","1","x1","0","0","1","1","x1^2 - x1","0","x1^2 - x1","1"],["0","1","x1^3 - 2*x1^2 + 4*x1 - 3","1","0","0","1","x1^4 - 3*x1^3 + 6*x1^2 - 7*x1 + 3","1","-x1^2 + 2*x1 - 3","x1^2 - x1 + 1"],["0","0","0","0","1","1","x1","-x1^3 + 2*x1^2 - 3*x1","x1^2 - x1 + 2","-x1^3 + 2*x1^2 - 3*x1","x1"]]}}