Lattices with a circulant generator matrix represent a subclass of cyclic lattices. This subclass can be described by a basis containing a vector and its circular shifts. In this paper, we present certain conditions under which the norm expression of an arbitrary vector of this type of lattice is substantially simplified, and then investigate some of the lattices obtained under these conditions. We exhibit systems of nonlinear equations whose solutions yield lattices as dense as $D_n$ in odd dimensions. As far as even dimensions, we obtain lattices denser than $A_n$ as long as $n \in 2\mathbb{Z} \backslash 4\mathbb{Z}$.