{"rank":4,"n":9,"seq":1134,"label":"4.9.1134","id":"r4_n9_0********0**********0**********************0***********************0********************0**************0****0************0****","beta_invariant":26,"birational_type":"Calabi-Yau threefold (elliptic over P^1 x P^1, K trivial; kappa = 0, not rational)","birational_type_components":"[{\"dim\":3,\"free_rank\":0,\"torus_rank\":0,\"core_dim\":3,\"qbar_components\":1,\"type\":\"Calabi-Yau threefold (elliptic over P^1 x P^1, K trivial; kappa = 0, not rational)\",\"reason\":\"genus-1 fibration over P^1 x P^1 with a section; K_B + L = O(0,0) - (0 exceptional)\",\"reduction_steps\":[\"double cover\"],\"elliptic_threefold\":{\"base\":\"P^1 x P^1\",\"N\":[2,2],\"canonical_drop\":0,\"canonical_class\":\"pi^* O(0, 0)\",\"discriminant_degrees\":[19,19],\"j_constant\":false},\"core_vars\":[\"x1\",\"x2\",\"x3\",\"x4\"],\"core_gens\":[\"x1^2*x2*x3^2*x4 - x1*x2^2*x3^2*x4 - x1^2*x2*x3*x4^2 + x1*x2^2*x3*x4^2 + x1^2*x2*x3^2 - x1^2*x2*x3*x4 - x1*x2*x3^2*x4 + x1^2*x3*x4^2 - x1*x2*x3*x4^2 + x2^2*x3*x4^2 + x1*x2*x4^3 - x2^2*x4^3 - x1*x3*x4^3 + x2*x3*x4^3 - x1^2*x3^2 + x1*x2*x3*x4 + x1*x3^2*x4 - x2*x3*x4^2\"]}]","birational_type_method":"threefold_elliptic_canonical_class","char_poly_splits":false,"char_set":"[]","expected_dimension_over_Z":null,"free_realization_space":false,"girth":4,"good_basis":null,"good_basis_v2":"1,3,8,9","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":false,"is_realizable_char0":false,"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":null,"is_transversal":false,"is_uniform":null,"n_automorphisms":9,"n_bases":117,"n_circuits":90,"n_components":null,"n_connected_components":1,"n_cyclic_flats":11,"n_flats":104,"n_loops":0,"principal_ideal":true,"rs_characteristic_dimension_unexpected":null,"rs_characteristic_dimension_varies":null,"rs_characteristic_dimensions":null,"rs_characteristic_dimensions_method":null,"rs_dim":-1,"rs_is_regular_scheme":null,"rs_is_smooth_over_QQ":null,"rs_is_smooth_over_ZZ":null,"rs_n_qbar_components":null,"rs_regularity_method":null,"rs_regularity_witness":null,"rs_scheme_simple_core_id":null,"rs_singular_fiber_characteristics":null,"rs_singular_fiber_characteristics_method":null,"rs_singular_fiber_primes":null,"rs_singular_fiber_primes_method":null,"rs_smoothness_method":null,"rs_smoothness_witness":null,"singular_primes":"[19]","smooth_char_0":true,"status":"computed_realization_space","tutte_polynomial":"x_0^4 + 5*x_0^3 + 15*x_0^2 + 9*x_0*x_1 + 26*x_0 + x_1^5 + 4*x_1^4 + 10*x_1^3 + 20*x_1^2 + 26*x_1","realization_space":{"vars":["x1","x2","x3","x4"],"ideal":["x1^2*x2*x3^2*x4 + x1^2*x2*x3^2 - x1^2*x2*x3*x4^2 - x1^2*x2*x3*x4 - x1^2*x3^2 + x1^2*x3*x4^2 - x1*x2^2*x3^2*x4 + x1*x2^2*x3*x4^2 - x1*x2*x3^2*x4 - x1*x2*x3*x4^2 + x1*x2*x3*x4 + x1*x2*x4^3 + x1*x3^2*x4 - x1*x3*x4^3 + x2^2*x3*x4^2 - x2^2*x4^3 + x2*x3*x4^3 - x2*x3*x4^2"],"inequations":["x3 - 1","x3 - x4","x1","x1*x3 - x2*x4","x1*x2*x3 - x1*x2*x4 + x1*x3*x4 - x1*x3 - x2*x3*x4 + x2*x4","x2 - x3","x2","x1 - 1","x2*x3 - x2*x4 + x3*x4 - x3","x1 - x2","x3","x1*x3 - x4","x1^2*x3 - x1*x2*x3 - x1*x3 + x2*x4","x1 - x4","x1*x2*x3 - x1*x3 - x2*x4 + x3*x4","x2 - 1","x4","x4 - 1","x2 - x4","x1*x2*x3^2 - x1*x2*x4^2 - x1*x3^2 + x1*x3*x4^2 - x2^2*x3*x4 + x2^2*x4^2 - x2*x3*x4^2 + x2*x3*x4","x1*x2*x3 - x1*x2*x4 - x1*x3 + x1*x4^2 - x2*x4^2 + x2*x4","x1*x3 - x1*x4^2 + x2*x4^2 - x2*x4","x1*x2*x3 - x1*x2*x4 - x1*x3*x4^2 + x1*x3*x4 - x1*x3 + x1*x4^2 + x2*x3*x4^2 - 2*x2*x3*x4 + x2*x4 + x3*x4 - x4^2","x1*x4 - x2*x4 + x2 - x4","x1^2*x2*x3^2 - x1^2*x2*x3*x4 + x1^2*x3^2*x4 - x1^2*x3^2 - 2*x1*x2*x3^2*x4 + x1*x2*x3*x4 + x1*x2*x4^3 + x1*x3^2*x4 - x1*x3*x4^3 + x2^2*x3*x4^2 - x2^2*x4^3 + x2*x3*x4^3 - x2*x3*x4^2","x1*x3 - x4^2"],"matrix":[["1","1","0","1","x2*x4 - x4","x1*x4 - x2*x4","1","0","0"],["0","1","1","0","x2*x4 - x3*x4","x1*x3 - x2*x4","1","0","0"],["0","1","0","x2","x1*x2*x3 - x1*x3","x1^2*x3 - x1*x2*x3","x1","1","0"],["0","1","0","x2","x2*x3*x4 - x3*x4","x1*x4^2 - x2*x4^2","x4","0","1"]]}}