Octic Hilbert 2-class fields of real quadratic fields with discriminant 8p

In this article we explain how to construct cyclic octic unramfied extensions of the real quadratic number field $k=\mathbb{Q}(\sqrt{2p})$, where $p\equiv 18$ is a prime number such that $h_2(k)\equiv 08$. The construction only requires solving the diophantine equation $e{u}^2=t^2+2p{s}^2$ in integers.