  When p divides the ordering of Galois group, the distribution of the Sylow
p-subgroup of Cl(K) is closely related to the problem of counting fields with
certain specifications. Moreover, different orderings of number fields affect
the answers of such questions in a nontrivial way. So, in this paper, we set up
an invariant of number fields with parameters, and consider field counting
problems with specifications while the parameters change as a variable. The
case of abelian extensions shows that the result of counting abelian fields has
a main term with parameters. The estimate of counting cubic fields with a
parameter shows that infinite moment is true for some ordering but not very
likely for the others.
