Lemma 6.1.
Assuming that:
char
K
≠
2
E
:
y
2
=
(
x
−
e
1
)
(
x
−
e
2
)
(
x
−
e
3
)
,
e
1
,
e
2
,
e
3
∈
K
distinct
Then
ω
=
d
x
y
is a differential on
E
with no zeroes or poles
⟹
g
(
E
)
=
1
. In particular, the
K
-vector space of regular differentials on
E
is 1-dimensional, spanned by
ω
.