Let the elliptic curves $E_{1}$ and $E_{2}$ be of the form $y^{2}=x^{3}+Ax+B$.
Addition between two points $P,Q\in E$ is defined as:
Suppose that there is a bijection from $E_{1}\to E_{2}$ for which points on $E_{1}$ are scaled by a factor of $\alpha$.
So the question is would this bijection preserve the property: $\alpha(P\oplus Q)=\alpha(P)\oplus\alpha(Q)$ for $P,Q\in E_{1}$ and $\alpha(P),\alpha(Q)\in E_{2}$?
The pdf that I'm reading derives the addition formula between two points on an elliptic curve to be:
For a secant line $y=\lambda x+v$ through $P_{1}=(x_{1},y_{1})$ and $P_{2}=(x_{2},y_{2})$: $$P_{1}\oplus P_{2}=(\lambda^2-x_{1}-x_{2},-\lambda^3+\lambda(x_{1}+x_{2})-v)$$ So it looks to be that the isomorphism property is not preserved according to the above formula, but can someone confirm it for me? I don't really have alot of experience with elliptic curves and group theory so I don't feel as though I am right.