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Let $k$ be a field of your choice— I'm particularly interested in algebraically closed fields. Are there explicit examples of curves over $k$ whose Jacobian is isogenous to the product of three copies of a CM elliptic curve $E$ over $k$? Yuri Zarhin's answer to When is a product of elliptic curves isogenous to the Jacobian of a hyperelliptic curve? says that when $k = \mathbb{C}$, such a curve exists for any $E$ (no CM assumption necessary).

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    $\begingroup$ There are a number of examples in this paper, §4. $\endgroup$
    – abx
    Commented Apr 19 at 3:36

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