  We find through a systematic analysis that all but 29,223 of the 473.8
million 4D reflexive polytopes found by Kreuzer and Skarke have a 2D reflexive
subpolytope. Such a subpolytope is generally associated with the presence of an
elliptic or genus one fibration in the corresponding birational equivalence
class of Calabi-Yau threefolds. This extends the growing body of evidence that
most Calabi-Yau threefolds have an elliptically fibered phase.
