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A prime number $p$ is said to be irregular if it divides the class number ofthe $p$-th cyclotomic field $\mathbb{Q}(\zeta_{p}) =\mathbb{Q}(\mathbb{G}_m[p])$. In this paper, we study its elliptic analogue forthe 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 classnumber of the $p^{n}$-division field $\mathbb{Q}(E[p^{n}])$ of an ellipticcurve $E$ of the form $y^{2} = x^{3}-(s^{4}+t^{2})x$. In particular, weconstruct a certain infinite subfamily consisting of curves with novelproperties 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 thatthese division fields are not isomorphic to each other. In our construction, weuse recent results obtained by the first author.

Elliptic analogue of irregular prime numbers for the $p^{n}$-division fields of the curves $y^{2} = x^{3}-(s^{4}+t^{2})x$