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

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This will be an incomplete answer as I don't know what Pic means. Also, I am not sure if what is below is the proper way to achieve what you want, but at least it may give you some first hints toward the good direction:

sage: K.<xi> = CyclotomicField(13)
sage: L.<t> = K[]
sage: R.<pinv> = L.extension(13*t-1)

This creates those rings:

sage: K
Cyclotomic Field of order 13 and degree 12
sage: L
Univariate Polynomial Ring in t over Cyclotomic Field of order 13 and degree 12
sage: R
Univariate Quotient Polynomial Ring in pinv over Univariate Polynomial Ring in t over Cyclotomic Field of order 13 and degree 12 with modulus 13*pinv - 1

And those elements:

sage: xi
xi
sage: pinv
1/13

which you can work with:

sage: xi^11
xi^11
sage: xi^12
-xi^11 - xi^10 - xi^9 - xi^8 - xi^7 - xi^6 - xi^5 - xi^4 - xi^3 - xi^2 - xi - 1
sage: pinv * xi^10
1/13*xi^10
sage: pinv * xi^10 + 14*pinv*xi^4+pinv^3
1/13*xi^10 + 14/13*xi^4 + 1/2197

Maybe there is a way to get that p instead of my pinv. I let someone else to give a better answer.