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We describe an algorithm to compute the reduction modulo $p$ of a crystallineGalois representation of dimension $2$ of$\text{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_p)$ with distinct Hodge-Tateweights via the semi-simple modulo $p$ Langlands correspondence. We give someexamples computed with an implementation of this algorithm in SAGE.

An algorithm for computing the reduction of $2$-dimensional crystalline representations of $\text{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_p)$