Consider a hyperelliptic curve $\mathcal{C}$ over $\mathbb{Q}$ and its Jacobian $J(\mathcal{C})$. Assume that $J(\mathcal{C})$ admits an elliptic factor $\mathcal{E}$.

For almost all primes, we can reduce $\mathcal{C}$ modulo $p$, and consider its zeta function $Z_p$. Can the elliptic factor property be seen as a special property of $Z_p$?

Conversly, if such property is satisfied for almost all primes, can we prove that a hyperelliptic curve possess an elliptic factor in its Jacobian?