International audienceLet E/Fq be an elliptic curve, and P a point in E(Fq) of prime order ℓ.Vélu's formulae let us compute a quotient curve E′=E/⟨P⟩ and rational maps defining a quotient isogeny ϕ:E→E′ in O~(ℓ)Fq-operations, where the O~ is uniform in q.This article shows how to compute E′, and ϕ(Q) for Q in E(Fq), using only O~(ℓ)Fq-operations, where the O~ is again uniform in q.As an application, this article speeds up some computations used in the isogeny-based cryptosystems CSIDH and CSURF