To let $ K $ to be a number field. To let $ K_ infty / K $ be a $ mathbb {Z} _p $ Extension. Iwasawa proved that there are four integers $ n_0, mu, lambda, nu geq 0 $ so for everyone $ n geq n_0 $, $$ mathrm {ord} _p (h_n) = mu p ^ n + lambda n + nu, $$

from where $ h_n $ is the class number of $ n $layer field $ K_n $ (so $ K_0 = K $).

My question is what we know about $ n_0 $? If we know how to calculate $ n_0 $then we can count $ mu, lambda, nu $ in finite steps, at least theoretically.

It's hard for me to find an explicit formula for $ n_0 $ due to the pseudo-isomorphism.

One may ask that this is the whole number $ r_0 $ so that $ K_ infty / K_ {r_0} $ is completely branched out with each branched main work? That's wrong. There are many examples in the work of Schoof-Kraft [Computing Iwasawa modules for real quadratic number fields], $ K_ infty / K_0 $ is completely branched at every branched prime, but $ n_0> 0 $,