Let $A$ be an abelian variety over a number field $k$ and let $NS_A$ denote its Neron-Severi scheme. Then the group of $k$-rational points of $NS_A$ is a finitely generated abelian group, i.e. $H^0(G_k,NS_A(\bar k)) = \mathbb{Z}^\rho \oplus Torsion$. I'd like to know:

What does a torsion element look like? Can it be ample? Is there an easy example, maybe of an elliptic curve, where the torsion of $NS_A$ is non-trivial?

Is the rank $\rho$ of $NS_A$, i.e. the Picard number, bounded by 1 and the square of the dimension of $A$?

Is the group of $\bar k$-rational points of $NS_A$ also a finitely generated abelian group?

Let $A$ be an abelian variety over a number field $k$ and let $NS_A$ denote its Neron-Severi scheme. Then the group of $k$-rational points of $NS_A$ is a finitely generated abelian group, i.e. $NS_A(k)=H^0(G_k,NS_A(\bar k)) = \mathbb{Z}^\rho \oplus Torsion$. I'd like to know:

What does a torsion element look like? Can it be ample? Is there an easy example, maybe of an elliptic curve, where the torsion of $NS_A(k)$ is non-trivial?

Is the rank $\rho$ of $NS_A(k)$, i.e. the Picard number, bounded by 1 and the square of the dimension of $A$?

Is the group of $\bar k$-rational points of $NS_A$ also a finitely generated abelian group?