Given an abelian variety X and a point a ∈ X we denote by < a > the closure of the subgroup of X generated by a. Let N = 2 g − 1. We denote by κ : X → κ(X) ⊂ P N the map from X to its Kummer variety. We prove that an indecomposable abelian variety X is the Jacobian of a curve if and only if there exists a point a = 2b ∈ X \ {0} such that < a > is irreducible and κ(b) is a flex of κ(X).