  The cosmic horseshoe gravitational lens is analyzed using the perturbative
approach. The two first order perturbative fields are expanded in Fourier
series. The source is reconstructed using a fine adaptive grid. The expansion
of the fields at order 2 produces a higher value of the chi-square. Expanding
at order 3 provides a very significant improvement, while order 4 does not
bring a significant improvement over order 3. The presence of the order 3 terms
is not a consequence of limiting the perturbative expansion to the first order.
The amplitude and signs of the third order terms are recovered by including the
contribution of the other group members. This analysis demonstrates that the
fine details of the potential of the lens could be recovered independently of
any assumptions by using the perturbative approach.
