What is the Frobenius element (if any) of this group.

Question

For $F = \mathbb{Q}[\sqrt{5}]$ with $p = 2$ and prime ideal over p of $ q = (2, 1 + \sqrt{5})$ with the Frobenius element defined as $$ x^{Frob_q} \equiv x^p (mod q) $$

with $Frob_q \in Gal (F/\mathbb{Q}) = C_2 $.