Embedding the Picard group inside the class group: the case of $\text{Q}$-factorial complete toric varieties

Let $X$ be a $\text{Q}$-factorial complete toric variety over an algebraic closed field of characteristic $0$. There is a canonical injection of the Picard group $\mathrm{P}\mathrm{i}\mathrm{c}(X)$ in the group $\mathrm{C}\mathrm{l}(X)$ of classes of Weil divisors. These two groups are finitely generated abelian groups; whilst the first one is a free group, the second one may have torsion. We investigate algebraic and geometrical conditions under which the image of $\mathrm{P}\mathrm{i}\mathrm{c}(X)$ in $\mathrm{C}\mathrm{l}(X)$ is contained in a free part of the latter group.