Coulangeon, R (reprint author), Univ Bordeaux 1, Inst Math Bordeaux, 351 Cours Liberat, F-334405 Talence, FranceIn this paper we compute gamma(K,2) for K = Q(rho), where rho is the real root of the polynomial x(3) - x(2) + 1 = 0. We refine some techniques introduced in [Baeza et al. 01] to construct all possible sets of minimal vectors for perfect forms. These refinements include a relation between minimal vectors and the Lenstra constant. This construction gives rise to results that can be applied in several other cases.