Let $X$ be a smooth, projective curve of genus $g \ge 2$. We know that the Jacobian $J(X)$ of the curve is a principally polarized abelian variety. The principal polarization is induced by the intersection form (or cup-product) on $H^1(X,\mathbb{Z})$. My question is: Is this the only principal polarization on $J(X)$? In other words, is there another unimodular, alternating, non-degenerate bilinear form on $H^1(X,\mathbb{Z})$ different from the cup-product (this will induce a different principal polarization on $J(X)$)?

I would think this is very basic, but I am not able find a good literature for this question. Any hint/reference will be most welcome.