Let $A$ be an abelian variety over a field $k$ of dimension $g\geq 2$.

There exists a finite morphism $A\to \mathbf{P}^g_k$. Here's the question.

Does there exist a finite morphism $A\to \mathbf{P}^g_k$ of degree two?

Can we say something about the minimal degree of a finite morphism $A\to \mathbf{P}^g_k$?