To handle the ever-increasing amount of data available, and to process it efficiently, querying data has become a ubiquitous task. In some cases, one also needs to be able to reason over data, by adding a logical layer called an ontology on top of data, giving rise to the problem of ontology-based query answering (OBQA). One way to express these ontologies is using existential rules, which are reasoning rules with the ability to create new individuals. A major drawback of existential rules is the undecidability of reasoning with them. One way of reasoning with existential rules is to rewrite the query you want to evaluate to take into account the rules. The usual algorithm rewrites conjunctive queries into union of conjunctive queries (UCQ), but several other target languages for rewritings have been studied. A natural choice for a target language is Datalog, a syntactic restriction of existential rules for which reasoning is decidable. Though, we show in this report that even when a set of existential rules admits a Datalog rewriting (we then say that it is Datalog-expressible), reasoning is still undecidable. This in particular means that, conversely to the well-studied case of UCQ-rewritings, Datalog-rewritings are not computable in general.