$E/\mathbb{C}$ and $E'/\mathbb{C}$ are isomorphic elliptic curves. Then if
$$E :\ y^{2} = x^3 + Ax + B$$
then
$$E': \ y^{2} = x^3 + \mu ^4 Ax + \mu ^6 B$$
and the isomorphism map $\phi : E \to E'$ is
$$\phi (x, y) = (\mu^2x, \mu^3y)$$
Except this isomorphism, is there any other isomorphism between the two curves $E$ and $E'$?
Best Answer
If you have two isomorphisms $E\to E'$ then you have an automorphism of $E$, those are found from the isomorphism $E\to \Bbb{C}/L,L=\Bbb{w_1Z+w_2Z}$, there is always $z\to -z$ which corresponds to $(x,y)\to (x,-y)$, there are more automorphisms only in two kind of elliptic curves with complex multiplication
either $L=r(\Bbb{Z+iZ})$ which corresponds to $j(E)= 1728$ and $B=0$ (the additional automorphism is $z\to iz,(x,y)\to (-x,iy)$)
or $L=r(\Bbb{Z+e^{2i\pi /3}Z})$ which corresponds to $j(E)=0,A=0$ (the additional automorphism is $z\to e^{2i\pi /3}z,(x,y)\to (e^{2i\pi /3}x,y)$)