Let $K$ be an algebraic number field and let $I$ be an ideal in $O_K$ (the ring of integers).

Denote by $d(I)$ the number of ideals that divide $I$. So if $I= \prod_{i=1}^k p_i^{e_i}$ is the decomposition to prime ideals, then $d(I) = (e_1+1)...(e_k+1)$.

I am looking for references for known bounds of $d(I)$ in terms of $N(I)$ for the general case or for the case $I$ is principal ideal.