  We study the Galois action attached to the Dwrok surfaces
$X_{\lambda}:X_0^4+X_1^4+X_2^4+X_3^4-4\lambda X_0X_1X_2X_3=0$ with parameter
$\lambda$ in a number field $F$. We show that when $X_{\lambda}$ has geometric
Picard number $19$, its N\'eron-Severi group $NS(\overline{X}_{\lambda})\otimes
\mathbb{Q}$ is a direct sum of quadratic characters. We provide two proofs to
this conclusion in our article. In particular, the geometrically proof
determines the conductor of each of quadratic characters. Our result matches
the one in \cite{Voight2}. With this decomposition, we give another proof to a
result of Wan.

Number Theory