Corollary 3.3. Assuming that:

$char K \neq 2, 3$ and $K$ is perfect
elliptic curves $E : y^{2} = x^{3} + a x + b$ , $E^{'} : y^{2} = x^{3} + a^{'} x + b^{'}$

Then

E

and

E^{'}

are isomorphic over

K

if and only if

a^{'} = u^{4} a

b^{'} = u^{6} b

for some

u \in K^{*}