  A prime number $p$ is said to be irregular if it divides the class number of
the $p$-th cyclotomic field $\mathbb{Q}(\zeta_{p}) =
\mathbb{Q}(\mathbb{G}_m[p])$. In this paper, we study its elliptic analogue for
the division fields of an elliptic curve. More precisely, for a prime number $p
\geq 5$ and a positive integer $n$, we study the $p$-divisibility of the class
number of the $p^{n}$-division field $\mathbb{Q}(E[p^{n}])$ of an elliptic
curve $E$ of the form $y^{2} = x^{3}-(s^{4}+t^{2})x$. In particular, we
construct a certain infinite subfamily consisting of curves with novel
properties that they are of Mordell-Weil rank 1 and the class numbers of their
$p^{n}$-division fields are divisible by $p^{2n}$. Moreover, we can prove that
these division fields are not isomorphic to each other. In our construction, we
use recent results obtained by the first author.
