We determine equations of the modular curves
X
1
(
N
)
{X_1}(N)
for
N
=
11
,
13
,
14
,
15
,
16
,
17
N = 11,13,14,15,16,17
and 18. Except for
N
=
17
N = 17
, these are the only existing elliptic or hyperelliptic
X
1
(
N
)
{X_1}(N)
. Applying these
X
1
(
N
)
{X_1}(N)
, we calculate tables of elliptic curves E over quadratic fields K with torsion groups of one of the following isomorphism types:
\[
E
tor
(
K
)
≅
Z
/
m
Z
,
m
=
11
,
13
,
14
,
15
,
16
and
18.
{E_{{\operatorname {tor}}}}(K) \cong {\mathbf {Z}}/m{\mathbf {Z}},\quad m = 11,13,14,15,16\;{\text {and}}\;18.
\]