Upper bound on smallest field norm divisible by p

Dec 2018
4
1
Earth
Let $K$ be a number field, $p$ a prime ideal, $N(p)$ the norm of $p$, $P$ a principal ideal and $N(P)$ the norm of $P$.

What are the (reasonable) upper bounds for the smallest $N(P)$ such that $N(p) | $N(P)$?

Thanks for help.