Let $C$ be a smooth projective curve of genus $2$ and $J$ denotes the Jacobian of $C$. Let $\theta$ be the image of $C$ under the abel Jacobi map. Is there exist a divisor $D$ in $J$ such that $D.\theta=1$?

Is it true that the condition $D.\theta=1$ implies that $J $ biregular to $ D\times \theta$?