Canonical divisor of an abelian variety

Let $A$ be an abelian variety over a field $k$ and let $K_A$ be its canonical divisor. Then I'm almost certain that $K_A$ is trivial, but I can't seem to prove it, nor find a counter example, nor find any reference on abelian varieties that even mentions canonical divisors.

Does anyone know a reference saying (or an idea for a short proof) that the canonical divisor on an abelian variety is trivial?

I'd prefer not to assume anything about $k$, but a result with $\operatorname{char}(k) = 0$ would be better than nothing.

-