Quintic reciprocity conjecture

This is a special case of a result due to Emma Lehmer (see this article). In (8), set $u = 0$, $v = 4t$ and $w = 4$ (observe that $x$, $u$ and $v$ are even, so $2$ is a quintic residue modulo $p$); then $16p = 16t^4 + 50 \cdot 16t^2 + 125 \cdot 16$, and division by $16$ produces the polynomial $p = t^4 + 50t^2 + 125$. I'm sure that a suitable choice of the parameters will produce your polynomial.