In Beauville's "Complex Algebraic Surfaces", given an elliptic surface $f : X \to C$ with a generic fiber $E$. Then either $\text{Alb}(X) \cong \text{Jac}(C)$ or there is an exact sequence of abelian varieties

$0 \to F \to \text{Alb}(X) \to \text{Jac}(C) \to 0$

with $F$ being isogenous to $E$.

In

Assume that $X$ is a properly elliptic that is in the second case, does . Does anyone know an example where $F$ is isomorphic to $E$ (and $X$ is not a product), and an example where $F$ is not isomorphic to $E$?

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# an exercise about elliptic surface in Beauville's book

In Beauville's "Complex Algebraic Surfaces", given an elliptic surface $f : X \to C$ with a generic fiber $E$. Then either $\text{Alb}(X) \cong \text{Jac}(C)$ or there is an exact sequence of abelian varieties

$0 \to F \to \text{Alb}(X) \to \text{Jac}(C) \to 0$

with $F$ being isogenous to $E$.

In the second case, does anyone know an example where $F$ is isomorphic to $E$ and an example where $F$ is not isomorphic to $E$?