  We prove that the sub-Riemannian exponential map is not injective in any
neighbourhood of certain critical points. Namely that it does not behave like
the injective map of reals given by $f(x) = x^3$ near its critical point $x =
0$. As a consequence, we characterise conjugate points in ideal sub-Riemannian
manifolds in terms of the metric structure of the space. The proof uses the
Hilbert invariant integral of the associated variational problem.
