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.
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