Transcendental lattices and supersingular reduction lattices of a singular $K3$ surface

A $K3$ surface $X$ defined over a field $k$ of characteristic 0 is called singular if the Néron-Severi lattice $\mathrm{NS}(X)$ of $X\otimes \overline{k}$ is of rank $20$. Let $X$ be a singular $K3$ surface defined over a number field $F$. For each embedding $\sigma: F\hookrightarrow \mathbb{C}$, we denote by $T(X^\sigma)$ the transcendental lattice of the complex $K3$ surface $X^\sigma$ obtained from $X$ by $\sigma$. For each prime $\mathfrak{p}$ of $F$ at which $X$ has a supersingular reduction $X_{\mathfrak{p}}$, we define $L(X, \mathfrak{p})$ to be the orthogonal complement of $\mathrm{NS}(X)$ in $\mathrm{NS}(X_{\mathfrak{p}})$. We investigate the relation between these lattices $T(X\sp\sigma)$ and $L(X,\mathfrak{p})$. As an application, we give a lower bound for the degree of a number field over which a singular $K3$ surface with a given transcendental lattice can be defined.