Let $S$ be a non-empty finite set of prime numbers, and let $F$ be an abelian extension over the rational field such that the Galois group of $F$ over some subfield of $F$ with finite degree is topologically isomorphic to the additive group of the direct product of the $p$-adic integer rings for all $p$ in $S$. Let $m$ be a positive integer that is neither congruent to $2$ modulo $4$ nor divisible by any prime number outside $S$ but divisible by all prime numbers in $S$. Let $\mathrm{\Omega }$ denote the composite of $p^n$-th cyclotomic fields for all $p$ in $S$ and all positive integers $n$. In our earlier paper [3], it is shown that there exist only finitely many prime numbers $l$ for which the $l$-class group of $F$ is nontrivial and the $m$-th cyclotomic field contains the decomposition field of $l$ in $\mathrm{\Omega }$. We shall prove more precise results providing us with an effective upper bound for a prime number $l$ such that the $l$-class group of $F$ is nontrivial and that the $m$-th cyclotomic field contains the decomposition field of $l$ in $\mathrm{\Omega }$.