Let $(E,O)$ be an elliptic curve. Let $\operatorname{Pic}^0(E)$ stand for the divisors that have degree $0$ where : $$D = \sum_{p\in E}n_p(P) \text{ and } \deg D = \sum_{p\in E}n_p.$$
I understand that the $\operatorname{Pic}^0(E)$ subgroup of the Picard group is isomorphic to the geometric group on the Elliptic curve. Is it true that the entire Picard group satisfies: $$\operatorname{Pic}(E) \cong \Bbb Z\oplus \operatorname{Pic}^0(E)?$$ I think the map should be something like: $$[D] \to (\deg D, [D] - (\deg D) [O])$$ where $[D]$ denotes the divisor class.
This is definitely a group homomorphism onto the first factor, however I don't believe it is one onto the second factor. Is it possible to modify the map/image somehow to get an isomorphism?