I am looking for some explicit examples of (elliptic) $K3$-families defined over a number field (better to be over $\mathbb{Q}$) with Picard number $18$ but does not admit Shioda-Inose structure, i.e. there is no primitive embedding from the transcendental lattice to $U^3$ ($U$ is the hyperbolic lattice). I am wondering is there good reference for that?
Sorry if the question is too trivial.