# How to Compute $\operatorname{Pic}(\mathbb{Z}[\zeta_p,1/p])$

Consider the extension $\mathbb{Q}(\zeta_p)/\mathbb{Q}$, where $\zeta_p$ is a primitive $p^{\text{th}}$ root of unity (and $p$ is a prime number). Form the ring of integers $\mathbb{Z}[\zeta_p]$. Now, invert the prime $p$, to obtain the ring $R=\mathbb{Z}[\zeta_p,1/p]$. I want to compute $\operatorname{Pic}(R)$.

Is this possible to do using sagemath? If so, how? How do I construct the ring $R$ using sagemath?

What's your definition of $\operatorname{Pic}$? You could construct $R$ as a quotient of $\mathbb{Z}[x,y]$.