Elliptic surfaces fibering over an elliptic curve with 12 singular fibers form a ten-dimensional moduli space. Their middle cohomology has a K3 type Hodge structure, endowing the moduli space with a period map into a ten-dimensional arithmetic quotient. I will discuss a proof that this period map is dominant (like for moduli of polarized K3 surfaces), but that the degree of the period map exceeds one (unlike the case for moduli of K3 surfaces).

This is joint work with F. Greer and A. Ward.