Rank of Mordell-Weil group and Euler characteristic of $\mathbb{G}_m$

Question

Given an elliptic curve $E$ over a number field $K$, let's choose a finite type model of $E$ over $\mathbb{Z}$ denoted by $E_{\mathbb{Z}}$. Why is the rank of Mordell-Weil group of $E$ or $\text{rank}(Pic^0(E(K)))$ equal to $\text{rank}(Pic(E_{\mathbb{Z}}))-\text{rank}(\mathcal{O}^{\times}(E_{\mathbb{Z}}))$? (the latter one is the rank of the invertible functions on $E_{\mathbb{Z}}$)