# Why are we not allowed to rescale the variables of the equation of an elliptic curve independently one from the other?

It seems that in whatever proof of the theorem that an elliptic curve can be put in Weierstrass form that you look at, the next step after getting an equation:

$$\alpha Y^2Z + a_1XY Z + a_3Y Z^2= \beta X^3+ a_2X^2Z + a_4XZ^2+ a_6Z^3$$

is to multiply through by $\beta^2/\alpha^3$ in order to make the linear change of variables $$\left(X\mapsto \frac\alpha\beta X, Y\mapsto \frac\alpha\beta Y\right)$$ to get rid of the leading coefficients. My question is: why are we restricted to linear change of variables that rescale $X$ and $Y$ in the same way (but still can send $Y\mapsto Y + mZ+nX$).

• Your question is not very clear. Are you asking if changes of variables of the form $X'=aX$, $Y'=bY$ are allowed? What would that accomplish? Feb 19, 2014 at 2:08
• @ÁlvaroLozano-Robledo yeah, I certainly expected that the change of variables $X'=\beta^{1/3}X, Y'=\sqrt{\alpha}Y$ would be allowed. I'd use that to get rid of the leading coefficients without touching on the $a_6$. But instead my reference insists that first one should multiply by $\beta^2/\alpha^3$ and then make the proportional change of variables. Feb 19, 2014 at 17:21

## 1 Answer

It is not that we can't rescale $X,Y$ independently, but rather that we don't want to rescale them by a square, or cubic root (as I suggested in my comment above $X'=\beta^{1/3}, Y'=\alpha^{1/2}Y$), which might not exist in the field $K$ over which the curve is defined.