Let $L$ be a cyclotomic field,  and $P$ a prime ideal of $\mathcal{O}_L$ 


1. is there any symbol for the equation $ax^2 + by^2 =1 \mod P$  and if so, is it computable in polynomial time?

2. is there any algorithm to find such $x$ and $y$ ? all such possible $(x,y)$?