Proposition 3.2.
Assuming that:
K
is perfect
E
,
E
′
are
elliptic curves
over
K
E
,
E
′
are in
Weierstrass form
Then
E
≅
E
′
over
K
if and only if
the equations are related by a change of variables
x
=
u
2
x
′
+
r
y
=
u
3
y
′
+
u
2
s
x
′
+
t
where
u
,
r
,
s
,
t
∈
K
with
u
≠
0
.