  In this article, we study a relation between certain quotients of ideal class
groups and the cyclotomic Iwasawa module $X_\infty$ of the Pontrjagin dual of
the fine Selmer group of an elliptic curve $E$ defined over $\mathbb{Q}$. We
consider the Galois extension field $K^E_n$ of $\mathbb{Q}$ generated by
coordinates of all $p^n$-torsion points of $E$, and introduce a quotient
$A^E_n$ of the $p$-sylow subgroup of the ideal class group of $K^E_n$ cut out
by the modulo $p^n$ Galois representation $E[p^n]$. We describe the asymptotic
behavior of $A^E_n$ by using the Iwasawa module $X_\infty$. In particular,
under certain conditions, we obtain an asymptotic formula as Iwasawa's class
number formula on the order of $A^E_n$ by using Iwasawa's invariants of
$X_\infty$.
