In this paper we introduce multiplicative lattices in and determine finite unions of suitable simplices as fundamental domains for sublattices of finite index. For this we define cyclic non-negative bases in arbitrary lattices. These bases are then used to calculate Shintani cones in totally real algebraic number fields. We mainly concentrate our considerations to lattices in two and three dimensions corresponding to cubic and quartic fields.
We present algorithms for the computation of extreme binary Humbert forms in real quadratic number fields. With these algorithms we are able to compute extreme Humbert forms for the number fields and . Finally we compute the Hermite-Humbert constant for the number field .
We present an algorithm for computing the 2-group of the positive divisor classes in case the number field has exceptional dyadic places. As an application, we compute the 2-rank of the wild kernel in .
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