Automorphisms of elliptic curve

Question

Consider an elliptic curve $y^2=x^3+b$ over $\mathbb{R}$. How to find all real automorphisms of this curve of order 3?

Answer 1

Let $E: y^2=x^3+Ax+B$ be an elliptic curve over $\mathbb{C}$. Any change of variables (from $E$ to another curve $E':Y^2=X^3+A'X+B'$ isomorphic to $E$ over $\mathbb{C}$) that preserves the (short) Weierstrass form is of the type $(u^2x,u^3y)=(X,Y)$. If this is an automorphism (i.e., $E=E'$) then we must have $u^4=1$ (if $A\neq 0$) and $u^6=1$ (if $B\neq 0$), because a change of variables $(u^2x,u^3y)=(X,Y)$ sends $E$ to $E':Y^2=X^3+u^4AX+u^6B$.

In particular, the automorphisms of $E:y^2=x^3+B$ are all of the form $(x,y)\mapsto (u^2x,u^3y)$ with $u^6=1$, so $u$ is a $6$th root of unity. Now you can easily check whether any of the automorphisms are defined over $\mathbb{R}$ and of order $3$.