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Let $E$ be an elliptic curve over $\mathbb{Q}$ which has multiplicativereduction at a fixed prime $p$. For each positive integer $n$ we put$K_n:=\mathbb{Q}(E[p^n])$. The aim of this paper is to extend the author'sprevious our results concerning with the order of the $p$-Sylow group of theideal class group of $K_n$ to more general setting. We also modify the previouslower bound of the order and describe the new lower bound in terms of theMordell-Weil rank of $E(\mathbb{Q})$ and the ramification related to $E$.

On the class numbers of the fields of the $p^n$-torsion points of elliptic curves over $\mathbb{Q}$