Explicit computations with crystalline cohomology

Question

I am currently studying crystalline cohomology and while all the talk about crystalline topoi is nice, I would like to see some explicit computations. What are some references on this subject which contain computations for smooth projective varieties over $\mathbb{F}_{p^n}$ or $\overline{\mathbb{F}_p}$ defined by explicit equations (with the action of Frobenius worked out too)? Or if you want, you can post an example of your own.