  We show that, for each $n>0$, there is a family of elliptic surfaces which
are covered by the square of a curve of genus $2n+1$, and whose Hodge
structures have an action by ${\mathbb Q}(\sqrt{-n})$. By considering the case
$n=3$, we show that one particular family of K3 surfaces are covered by the
square of genus $7$. Using this, we construct a correspondence between the
square of a curve of genus $7$ and a general K3 surface in ${\mathbb P}^4$ with
$15$ ordinary double points up to isogeny. This gives an explicit proof of the
Kuga-Satake-Deligne correspondence for these K3 surfaces and any K3 surfaces
isogenous to them, and further, a proof of the Hodge conjecture for the squares
of these surfaces. We conclude that the motives of these surfaces are
Kimura-finite. Our analysis gives a birational equivalence between a moduli
space of curves with additional data and the moduli space of these K3 surfaces
with a specific elliptic fibration.
