  We construct non-geometric string compactifications by using the F-theory
dual of the heterotic string compactified on a two-torus with two Wilson line
parameters, together with a close connection between modular forms and the
equations for certain K3 surfaces of Picard rank $16$. We construct explicit
Weierstrass models for all inequivalent Jacobian elliptic fibrations supported
on this family of K3 surfaces and express their parameters in terms of modular
forms generalizing Siegel modular forms. In this way, we find a complete list
of all dual non-geometric compactifications obtained by the partial higgsing of
the heterotic string gauge algebra using two Wilson line parameters.
