In [BGK+18] in section 4, Boneh et al. write that:

For any choice of ideal classes $\mathfrak{a}_1,\dots,\mathfrak{a}_n,\mathfrak{a}_1',\dots,\mathfrak{a}_n'$ in ${Cl}(\mathcal{O})$, the abelian varieties \begin{align} (\mathfrak{a}_1 \star E) \times \dots \times (\mathfrak{a}_n \star E) \text{ and } (\mathfrak{a}_1' \star E) \times \dots \times (\mathfrak{a}_n' \star E) \end{align} are isomorphic over $\mathbb{F}_q$ iff $\mathfrak{a}_1 \cdots \mathfrak{a}_n = \mathfrak{a}_1' \cdots \mathfrak{a}_n' $ in ${Cl}(\mathcal{O})$. In particular: \begin{align} (\mathfrak{a}_1 \star E) \times \dots \times (\mathfrak{a}_n \star E) \cong (\mathfrak{a}_1\cdots\mathfrak{a}_n) \star E \times E^{n-1} \end{align}

There is no proof in the paper and I did not succeed proving this myself. Can someone point me to a proof for these assertions?


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