The discriminant of an algebraic torus

For a torus T defined over a global field K, we revisit an analytic class number formula obtained by Shyr in the 1970s as a generalization of Dirichletʼs class number formula. We prove a local-global presentation of the quasi-discriminant of T, which enters into this formula, in terms of cocharacters of T. This presentation can serve as a more natural definition of this invariant.