Decomposition into isogeny factors (reviewed)

If an abelian variety $A$ is not simple, it is isogenous to a product of simple lower dimensional abelian varieties. These simple abelian varieties $B_i$ are the isogeny factors of $A$, and we say that $A$ decomposes (up to isogeny) into the product of the $B_i$'s: $$A \sim B_1 \times \cdots \times B_n$$ Note that two elements of this product might be isogenous; in other words, elements of the decomposition may appear with multiplicity.