  We find all smooth toric bases that support elliptically fibered Calabi-Yau
threefolds, using the intersection structure of the irreducible effective
divisors on the base. These bases can be used for F-theory constructions of
six-dimensional quantum supergravity theories. There are 61,539 distinct
possible toric bases. The associated 6D supergravity theories have a number of
tensor multiplets ranging from 0 to 193. For each base an explicit Weierstrass
parameterization can be determined in terms of the toric data. The toric
counting of parameters matches with the gravitational anomaly constraint on
massless fields. For bases associated with theories having a large number of
tensor multiplets, there is a large non-Higgsable gauge group containing
multiple irreducible gauge group factors, particularly those having algebras
e_8, f_4 and (g_2 + su(2)) with minimal (non-Higgsable) matter.
