It is a general fact that on any simple abelian variety, an effective divisor is ample. The following result underlies the usual algebraic proof of the projectivity of abelian varieties (defined initially only as complete group varieties). It is therefore rather standard; the first reference which comes to mind is Lemma 8.5.6 on page 253 in the abelian varieties chapter of the book Heights in diophantine geometry by Bombieri and Gubler, from which I quote literally.
Let A be an abelian variety and $D$ an effective divisor such that the subgroup $$ Z_D : \hspace{3cm} \{ a \in A \mid a + D = D \} $$ is finite. Then $D$ is ample on $A$.
If the abelian variety $A$ is simple, and $D$ is non-zero, then the $Z_D$ is a fortiori finite, since it is a proper algebraic subgroup of $A$; and it follows from the quoted Lemma 8.5.6 that $D$ is ample.
An application. A surjective morphism $f: A \to X$ from a simple abelian variety onto a positive-dimensional projective variety $X$ is finite.
Proof. Choose $H \subset X$ an ample divisor. The divisor $f^*H$ is effective on $A$, hence it is ample. This is equivalent to $f$ being finite.