### solving system of equations over number field modulo another number field

I am trying to solve two, 2-variable polynomial equations over $F:=\mathbb{Q}(i)$ modulo $K:=F(\sqrt{2})$.
Specifically, if p1 = $a^2+6b^2$, p2 = $3a^2+2b^2$, and ~~$K~~*^{4}$, **$K^{4}$*

* **i.e. the group of 4th powers of nonzero elements of $K$. I want to find (all?) $a$ and $b$ in $F$ such that p1$\equiv$1 modulo $K^{**4}$ and p2$\equiv$-1 modulo *~~$K^{*4}$.~~$K^{4}$.

Any amount of walk through or pointing in the right direction, or telling me this might not be doable would be great! I am relatively new to sage, or at least it has been years since I've used it.