  The degree of a curve $C$ in a polarized abelian variety $(X,\lambda)$ is the
integer $d=C\cdot\lambda$. When $C$ generates $X$, we find a lower bound on $d$
which depends on $n$ and the degree of the polarization $\lambda$. The smallest
possible degree is $d=n$ and is obtained only for a smooth curve in its
Jacobian with its principal polarization (Ran, Collino). The cases $d=n+1$ and
$d=n+2$ are studied. Moreover, when $X$ is simple, it is shown, using results
of Smyth on the trace of totally positive algebraic integers, that if $d\le
1.7719\, n$, then $C$ is smooth and $X$ is isomorphic to its Jacobian. We also
get an upper bound on the geometric genus of $C$ in terms of its degree.
