Norm one tori and Hasse norm principle

Let $k$ be a field and $T$ be an algebraic $k$-torus. In 1969, over a global field $k$, Voskresenskii proved that there exists an exact sequence $0\to A(T)\to H^1(k,{\rm Pic},\overline{X})^\vee\to Sha(T)\to 0$where$A(T)$ is the kernel of the weak approximation of $T$, $Sha(T)$ is the Shafarevich-Tate group of $T$, $X$ is a smooth $k$-compactification of $T$, $\bar{X}=X{\times }_k\bar{k}$, $\mathrm{P}\mathrm{i}\mathrm{c}\bar{X}$ is the Picard group of $\bar{X}$ and $\vee$ stands for the Pontryagin dual. On the other hand, in 1963, Ono proved that for the norm one torus $T=R_{K\mathrm{/}k}^{(1)}(G_m)$ of $K\mathrm{/}k$, $Sha(T)=0$ if and only if the Hasse norm principle holds for $K\mathrm{/}k$. First, we determine $H^1(k,\mathrm{P}\mathrm{i}\mathrm{c}\bar{X})$ for algebraic $k$-tori $T$ up to dimension $5$. Second, we determine $H^1(k,\mathrm{P}\mathrm{i}\mathrm{c}\bar{X})$ for norm one tori $T=R_{K\mathrm{/}k}^{(1)}(G_m)$ with $[K:k]=n\le 15$ and $n\mathrm{\ne }12$. We also show that $H^1(k,\mathrm{P}\mathrm{i}\mathrm{c}\bar{X})=0$ for $T=R_{K\mathrm{/}k}^{(1)}(G_m)$ when the Galois group of the Galois closure of $K\mathrm{/}k$ is the Mathieu group $M_n\le S_n$ with $n=11,12,22,23,24$. Third, we give a necessary and sufficient condition for the Hasse norm principle for $K\mathrm{/}k$ with $[K:k]=n\le 15$ and $n\mathrm{\ne }12$. As applications of the results, we get the group $T(k)\mathrm{/}R$ of $R$-equivalence classes over a local field $k$ via Colliot-Th'{e}l`{e}ne and Sansuc’s formula and the Tamagawa number $\tau (T)$ over a number field $k$ via Ono’s formula $\tau (T)=\mathrm{\mid }{H}^1(k,\widehat{T})\mathrm{\mid }\mathrm{/}\mathrm{\mid }Sha(T)\mathrm{\mid }$.