Let $k$ be a field, $T$ be an algebraic $k$-torus, $X$ be a smooth $k$-compactification of $T$ and ${ \rm Pic}\,\overline{ X}$ be the Picard group of $\overline{ X}=X\times_k\overline{ k}$. Hoshi, Kanai and Yamasaki [HKY22] determined $H^1(k,{ \rm Pic}\, \overline{ X})$ for norm one tori $T=R^{ (1)}_{ K/k}(G_m)$ and gave a necessary and sufficient condition for the Hasse norm principle for extensions $K/k$ of number fields with $[K:k]=n\leq 15$ and $n\neq 12$. In this paper, we determine $64$ cases with $H^1(k,{ \rm Pic}\, \overline{ X})\neq 0$ and give a necessary and sufficient condition for the Hasse norm principle for $K/k$ where $[K:k]=12$.
Categories: math.AG math.NT