Episturmian morphisms generalize Sturmian morphisms. They are definedas compositions of exchange morphisms and two particular morphismsL, and R. Epistandard morphisms are the morphisms obtained withoutconsidering R. In [14], a general study of these morphimsand of conjugacy of morphisms is given. Here, given a decomposition of an Episturmian morphism f over exchange morphisms and {L,R},we consider two problems: how to computea decomposition of one conjugate of f;how to compute a list of decompositions of all the conjugates of f when f is epistandard.For each problem, we give several algorithms.Although the proposed methods are fundamently different, we show thatsome of these lead to the same result. We also give other algorithms, using the same input, to compute for instance the length of the morphism, or its number of conjugates.