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 $E$?
