  We prove that there are finitely many families, up to isomorphism in
codimension one, of elliptic Calabi-Yau manifolds $Y\rightarrow X$ with a
rational section, provided that $\dim(Y)\leq 5$ and $Y$ is not of product-type.
As a consequence, we obtain that there are finitely many possibilities for the
Hodge diamond of such manifolds. The result follows from log birational
boundedness of klt pairs $(X, \Delta)$ with $K_X+\Delta$ numerically trivial
and not of product-type, in dimension at most $4$.
