  In this note, we describe an infinite family of sharp Szemer\'{e}di-Trotter
constructions. These constructions are cartesian products of arbitrarily high
dimensional generalized arithmetic progressions (GAPs), where the bases for
these GAPs come from arbitrary number fields over $\mathbb{Q}$. This can be
seen as an extension of a recent result of Guth and Silier, who provided
similar constructions based on the field $\mathbb{Q}(\sqrt{k})$ for square-free
$k$. However, our argument borrows from an idea of Elekes, which produces
cartesian products where the parts are of unequal size. This significantly
simplifies the analysis and allows us to easily give constructions coming from
any number field.
