In his work on mirror symmetry (http://arxiv.org/pdf/alg-geom/9502005v2.pdf) Igor Dolgachev has considered families of K3 surfaces of Picard rank at least 19 with the base given by $X_0(n)^+$, the quotient of the modular curve of level $n$ by Fricke involution.

Consider a modular parametrization $\mu:X_0(n)^+ \to E$ for some rational elliptic curve $E$, say of rank two. A rational point $p$ on $E$ gives rise to a collection of points $\mu^{-1}(p)$ on $X_0(n)^+$ and therefore to a finite set of the aforementioned K3 surfaces.

Question: What is special about these K3 surfaces? For example, is their Picard group of rank 19 or does it jump to 20? Do they have some other algebra-geometric characterization, such as existence of some sections of some line bundle on them? Has this been studied at all?

P.S. My guess is that the jump in the rank of K3 would be some CM condition, and thus not terribly interesting. But then what is special about these K3-s that correspond to points of rank 2 curve?