Let $T$ be an algebraic torus over a number field $K$. 
Following notations in Ono's The Arithmetic of Tori, 
http://www.jstor.org/discover/10.2307/1970307?sid=21105671135711&uid=3739888&uid=2&uid=3739256&uid=4

we define 
$T_A$ the adele points, $T_K$ the $K$-rational point of $T$ and 
$$T_{A,S_{\infty}}=\prod_{v \in S_{\infty}}T(K_v)\times \prod_{v \notin S_{\infty}}T_v^c $$
The class number of $T$ is defined as 
$$ h_T := [{T_A}:{T_{A,S_{\infty}}T_K}] $$
My question is how do we compute $h_T$ in pratice. If $T=\mathbb{G}_m$, $h_T$ is just the class number of $K$ and we compute it using Minkowski's bounds. But I don't know how to compute $h_T$ in general. 

For an example, if $T$ is the torus $Spec(\mathbb{Q}[x,y]/(x^2+y^2-1))$, what is $h_T$ ?
Thank you very much.