Let $C$ be a general genus $g$ curve, how can we describe the Neron-Severi group of its $n$-th self product $C^n=C\times \dots \times C$?

It is a lattice in $H^2(C^n,\mathbb{Z})\cong \mathbb{Z}^{n+2n(n-1)g^2}$, do we know its rank and basis if $C$ is general?