We prove detailed asymptotics for the number of spanning trees, called complexity, for a general class of discrete tori as the parame- ters tend to infinity. The proof uses in particular certain ideas and techniques from an earlier paper [CJK10]. Our asymptotic formula provides a link between the complexity of these graphs and the height of associated real tori, and allows us to deduce some corollaries on the complexity thanks to certain results from analytic number theory. In this way we obtain a conjectural relationship between complexity and regular sphere packings.