Ideal class groups of number fields and Bloch-Kato's Tate-Shafarevich groups for symmetric powers of elliptic curves

For an elliptic curve $E$ over $\mathbb{Q}$, putting $K=\mathbb{Q}(E[p])$ which is the $p$-th division field of $E$ for an odd prime $p$, we study the ideal class group ${\mathrm{C}\mathrm{l}}_K$ of $K$ as a $\mathrm{G}\mathrm{a}\mathrm{l}(K\mathrm{/}\mathbb{Q})$-module. More precisely, for any $j$ with $1⩽j⩽p-2$, we give a condition that $$\mathrm{C}\mathrm{l}$_K\otimes $\mathbb{F}$_p$hasthesymmetricpower$$\mathrm{S}\mathrm{y}\mathrm{m}$^j E[p]$of$E[p]$ as its quotient $\mathrm{G}\mathrm{a}\mathrm{l}(K\mathrm{/}\mathbb{Q})$-module, in terms of Bloch-Kato’s Tate-Shafarevich group of ${\mathrm{S}\mathrm{y}\mathrm{m}}^j{V}_{p}E$. Here $V_{p}E$ denotes the rational $p$-adic Tate module of $E$. This is a partial generalization of a result of Prasad and Shekhar for the case $j=1$.