$\newcommand{\cris}{\mathrm{cris}}$In my setting, $K/\mathbb Q_p$ is finite and unramified, and $V$ is a $2$-dimensional crystalline representation of $G_K$. Then we have $D_{\cris}(V)$, which is $2$-dim filtered $\phi$ module. I need to calculate a basis for $D_{\cris}(V)$, the matrix of $\phi$ and the filtration. (p-adic approximate would also be of help).
I wonder if there exists a effective way to calculate $D_{\cris}$ for general Galois representations? Since the only few examples (of calculating $D_{\cris}$) I know are characters and Tate module which are deeply dependent on the structure of the representations and the methods seems not generalizable.
To respond Sawin: The Galois representation I care about is quite general, it comes from a series of lifting $\mathbb Z/p^n$ representations. It's true that every Galois rep contains infinite data, but the data can be recovered by looking at mod $p^n$ representations step by step. So I wonder if there is a canonical way to use the reduced representations $\rho_n:G_K\to GL_n(\mathbb Z/p^n)$ to calculate the filtered $\phi$-module.
