Lenstra's Constant and Extreme Forms in Number Fields

Abstract

In this paper we compute $\gamma_{K,2$ for $K=\mathbb{Q}(\rho)$, where $\rho$ is the real root of the polynomial $x^3 -x^2 +1 =0$. We refine some techniques introduced in Baeza, et al. 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.