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 $$ h_T := \frac{T_A}{T_{A,S_{\infty}}T_K} $$ 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.

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.