Let $P$ be a polytope in $\Bbb Q^d$ and let $c$ be the smallest positive integer so that $c\cdot P$ has vertices in $\Bbb Z^d$.
Is there standard vocabulary to refer to $c$? Something like the integral index of $P$ makes sense, but I haven't seen terms like this used before.
A direct quote from "The minimum period of the Ehrhart quasi-polynomial of a rational polytope" by Tyrrell B. McAllister and Kevin M. Woods:
Given a rational polytope $P \subset \mathbb{R}^{d}$, the denominator of $P$ is $$\mathcal{D}(P)=\min \left\{n \in \mathbb{Z}_{>0}: n P \text { is an integral polytope }\right\}.$$