Let be a field and be an algebraic -torus. In 1969, over a global field , Voskresenskiǐ proved that there exists an exact sequence where is the kernel of the weak approximation of , is the Shafarevich-Tate group of , is a smooth -compactification of , , is the Picard group of and stands for the Pontryagin dual. On the other hand, in 1963, Ono proved that for the norm one torus of , if and only if the Hasse norm principle holds for . First, we determine for algebraic -tori up to dimension . Second, we determine for norm one tori with and . We also show that for when the Galois group of the Galois closure of is the Mathieu group with . Third, we give a necessary and sufficient condition for the Hasse norm principle for with and . As applications of the results, we get the group of -equivalence classes over a local field via Colliot-Thélène and Sansuc’s formula and the Tamagawa number over a number field via Ono’s formula .