2 Changed $\mathbb{Z}^5$ to $\mathbb{Z}^6$, and dimension 5 to dimension 6
First, let $A$ be a generic abelian surface (generic abelian variety of dimension 2) over $\mathbb{C}$. Then the group of cycles of dimension 1 (divisors) up to rational equivalence is known, it its the Picard group of $A$, see e.g. the answers to this question and Fulton, Intersection Theory, Chapter 19. Still I have a stupid question: Can one "write down" all the (positive?) divisors on $A$?