  Let $M$ be the moduli space of rank $2$ stable bundles with fixed determinant
of degree $1$ on a smooth projective curve $C$ of genus $g\ge 2$. When $C$ is
generic, we show that any elliptic curve on $M$ has degree (respect to
anti-canonical divisor $-K_M$) at least 6, and we give a complete
classification for elliptic curves of degree $6$. Moreover, if $g>4$, we show
that any elliptic curve passing through the generic point of $M$ has degree at
least $12$. We also formulate a conjecture for higher rank.
