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Is an abelian surface containing an elliptic curve a bielliptic surface?

Suppose I have an abelian surface $A$ over the complex numbers that contains an elliptic curve $E$. Then $A \to A/E$ is an elliptic fibration. The abelian surface $A$ will also contain a complementary elliptic curve $F$. Then $A \to A/F$ should also define an elliptic fibration.

Does this make A bielliptic? And does this always hold so long as it contains one elliptic curve?

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    $\begingroup$ What is the definition of bielliptic? $\endgroup$
    – Yosemite Stan
    Commented Feb 6 at 1:12

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By definition https://en.wikipedia.org/wiki/Hyperelliptic_surface the Albanese morphism of a bielliptic surface has 1-dimensional fibers, while the Albanese morphism of an abelian surface is the identity. So, no, an abelian surface is never bielliptic.

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  • $\begingroup$ Thank you very much! $\endgroup$
    – Stormblessed
    Commented Feb 6 at 12:27

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