  We present a formula for the class number of a multinorm one torus $T_{L/k}$
associated to any \'etale algebra $L$ over a global field $k$. This is deduced
from a formula for analogues of invariants introduced by T.~Ono, which are
interpreted as a generalization of Gauss genus theory. This paper includes the
variants of Ono's invariant for arbitrary $S$-ideal class numbers and the
narrow version, generalizing results of Katayama, Morishita, Sasaki and Ono.
