Let
k
k
be a field and
T
T
be an algebraic
k
k
-torus. In 1969, over a global field
k
k
, Voskresenskiǐ proved that there exists an exact sequence
0
→
A
(
T
)
→
H
1
(
k
,
Pic
X
¯
)
∨
→
Ш
(
T
)
→
0
0\to A(T)\to H^1(k,\operatorname {Pic}\overline {X})^\vee \to \Sha (T)\to 0
where
A
(
T
)
A(T)
is the kernel of the weak approximation of
T
T
,
Ш
(
T
)
\Sha (T)
is the Shafarevich-Tate group of
T
T
,
X
X
is a smooth
k
k
-compactification of
T
T
,
X
¯
=
X
×
k
k
¯
\overline {X}=X\times _k\overline {k}
,
Pic
X
¯
\operatorname {Pic}\overline {X}
is the Picard group of
X
¯
\overline {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
/
k
(
1
)
(
G
m
)
T=R^{(1)}_{K/k}(\mathbb {G}_m)
of
K
/
k
K/k
,
Ш
(
T
)
=
0
\Sha (T)=0
if and only if the Hasse norm principle holds for
K
/
k
K/k
. First, we determine
H
1
(
k
,
Pic
X
¯
)
H^1(k,\operatorname {Pic} \overline {X})
for algebraic
k
k
-tori
T
T
up to dimension
5
5
. Second, we determine
H
1
(
k
,
Pic
X
¯
)
H^1(k,\operatorname {Pic} \overline {X})
for norm one tori
T
=
R
K
/
k
(
1
)
(
G
m
)
T=R^{(1)}_{K/k}(\mathbb {G}_m)
with
[
K
:
k
]
=
n
≤
15
[K:k]=n\leq 15
and
n
≠
12
n\neq 12
. We also show that
H
1
(
k
,
Pic
X
¯
)
=
0
H^1(k,\operatorname {Pic} \overline {X})=0
for
T
=
R
K
/
k
(
1
)
(
G
m
)
T=R^{(1)}_{K/k}(\mathbb {G}_m)
when the Galois group of the Galois closure of
K
/
k
K/k
is the Mathieu group
M
n
≤
S
n
M_n\leq S_n
with
n
=
11
,
12
,
22
,
23
,
24
n=11,12,22,23,24
. Third, we give a necessary and sufficient condition for the Hasse norm principle for
K
/
k
K/k
with
[
K
:
k
]
=
n
≤
15
[K:k]=n\leq 15
and
n
≠
12
n\neq 12
. As applications of the results, we get the group
T
(
k
)
/
R
T(k)/R
of
R
R
-equivalence classes over a local field
k
k
via Colliot-Thélène and Sansuc’s formula and the Tamagawa number
τ
(
T
)
\tau (T)
over a number field
k
k
via Ono’s formula
τ
(
T
)
=
|
H
1
(
k
,
T
^
)
|
/
|
Ш
(
T
)
|
\tau (T)=|H^1(k,\widehat {T})|/|\Sha (T)|
.