Hensel-lifting torsion points on Jacobians and Galois representations

Let $\rho$ be a mod $\ell$ Galois representation. We show how to compute $\rho$, given the characteristic polynomial of the image of the Frobenius at one prime $p$ and a curve $C$ whose Jacobian contains $\rho$ in its $\ell$-torsion. The main ingredient is a method to $p$-adically lift torsion points on a Jacobian in the framework of Makdisi's algorithms.