Mordell-Weil groups of the Jacobian of the 5-th Fermat curve

Let $J_{5}$ denote the Jacobian of the Fermat curve of exponent 5 and let $K=Q(\zeta _{5})$. We compute the groups $J_{5}(K)$, $J_{5}(K^{+})$, $J_{5}(Q)$, where $K^{+}$ is the unique quadratic subfield of $K$. As an application, we present a new proof that there are no $K$-rational points on the 5-th Fermat curve, except the so called ``points at infinity".