In Lectures on Vector Bundles, Le Potier states that for an elliptic curve $X$ we have $$ X \cong \operatorname{Pic}^1(X).$$ As far as I understand the map $x \mapsto [x]-[O]$ gives an isomorphism of $X$ and $\operatorname{Pic}^0(X)$, where $O$ is the point at infinity of the elliptic curve.
Why does the isomorphism above exist and why is it not a contradiction to $X \cong \operatorname{Pic}^0(X)$?