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The paper of Hayashida and Nishi "Existence of curves of genus two in the product of smooth elliptic curves", J. Math. Soc. Jap 17, partially answers the question:

"When is the product of two elliptic curves $isomorphic$ to the Jacobian of a genus $2$ curve?"

The result is the following:

THEOREM Consider a product of two elliptic curves $A:=E \times F$, whose ring of endomorphism is isomorphic to the principal order of an imaginary quadratic field $Q(\surd-m)$. Then $A$ can be a Jacobian for all values of $m$ except $1, 3, 7$ and $15$. Moreover, there are only finitely many curves of genus $2$ on $A$ up to isomorphism.

2 added 5 characters in body

The paper of Hayashida and Nishi "Existence of curves of genus two in the product of smooth elliptic curves", J. Math. Soc. Jap 17, answers the question:

"When is the product of two elliptic curves $isomorphic$ to the Jacobian of a genus $2$ curve?curve?"

The results result is the following:

THEOREM Consider a product of two elliptic curves $A:=E \times F$, whose ring of endomorphism is isomorphic to the principal order of an imaginary quadratic field $Q(\surd-m)$. Then $A$ can be a Jacobian for all values of $m$ except $1, 3, 7$ and $15$. Moreover, there are only finitely many curves of genus $2$ on $A$ up to isomorphism.

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The paper of Hayashida and Nishi "Existence of curves of genus two in the product of smooth elliptic curves", J. Math. Soc. Jap 17 answers the question:

"When is the product of two elliptic curves $isomorphic$ to the Jacobian of a genus $2$ curve?

The results is the following:

THEOREM Consider a product of elliptic curves $A:=E \times F$, whose ring of endomorphism is isomorphic to the principal order of an imaginary quadratic field $Q(\surd-m)$. Then $A$ can be a Jacobian for all values of $m$ except $1, 3, 7$ and $15$. Moreover, there are only finitely many curves of genus $2$ on $A$ up to isomorphism.