  Let $\mathbb{P}$ denote the weighted projective space with weights
$(1,1,1,3)$ over the rationals, with coordinates $x,y,z,$ and $w$; let
$\mathcal{X}$ be the generic element of the family of surfaces in $\mathbb{P}$
given by \begin{equation*}
  X\colon w^2=x^6+y^6+z^6+tx^2y^2z^2. \end{equation*} The surface $\mathcal{X}$
is a K3 surface over the function field $\mathbb{Q}(t)$. In this paper, we
explicitly compute the geometric Picard lattice of $\mathcal{X}$, together with
its Galois module structure, as well as derive more results on the arithmetic
of $\mathcal{X}$ and other elements of the family $X$.
