  An algorithm to systematically construct all Calabi-Yau elliptic fibrations
realized as hypersurfaces in a toric ambient space for a given base and gauge
group is described. This general method is applied to the particular question
of constructing SU(5) GUTs with multiple U(1) gauge factors. The basic data
consists of a top over each toric divisor in the base together with
compactification data giving the embedding into a reflexive polytope. The
allowed choices of compactification data are integral points in an auxiliary
polytope. In order to ensure the existence of a low-energy gauge theory, the
elliptic fibration must be flat, which is reformulated into conditions on the
top and its embedding. In particular, flatness of SU(5) fourfolds imposes
additional linear constraints on the auxiliary polytope of compactifications,
and is therefore non-generic. Abelian gauge symmetries arising in toric
F-theory compactifications are studied systematically. Associated to each top,
the toric Mordell-Weil group determining the minimal number of U(1) factors is
computed. Furthermore, all SU(5)-tops and their splitting types are determined
and used to infer the pattern of U(1) matter charges.
