Natural classes in cohomology ring of symmetric product of curves

Question

Let $X$ be a smooth projective algebraic curve and let $X^{(n)}$ be the $n$-fold symmetric product. We consider elements of $X^{(n)}$ as effective divisors on $X$ of degree $n$. At least when $n$ is large compared to the genus of $X$, the cohomology ring of $X^{(n)}$ can be understood very well since $X^{(n)}$ can be realized as a projective space bundle over the Jacobian of $X$. More precisely, over the cohomology ring of the Jacobian it is generated by the class of $X_P$. Here $X_P$ is the set of all divisors containing $P$ (as on page 590 of "3264" by Eisenbud and Harris). For any partition $(d_1,\ldots,d_r)$ of $n$ we have the following map $$X\times\cdots\times X\to X^{(n)},\,\, (P_1,\ldots,P_r)\mapsto d_1P_1+\ldots+d_rP_r.$$ How can I see which cohomology class has the image of this map?