**Elliptic K3 surfaces.** Let $X$ be a general projective elliptic K3 of Picard rank two. Assume that singular fibers of the elliptic fibrations are of type $A_1$ so that by the Euler characteristic count there are $e(X) = 24$ singular fibers. I do not require existence of a section in the definition of an elliptic K3 surface.

**Question**: do these $24$ points determine the general such elliptic K3 uniquely, or at least up to finitely many choices? If not, what can be said about the dimension of the moduli space of those elliptic K3 surfaces with a prescribed set of $24$ singular fibers?

Any suggestions or references welcome.