Say we have an elliptic curve in its most general form:
$Ax^3 + Bx^2 y + Cxy^2 + Dy^3 + Ex^2 + Fxy + Gy^2 + Hx + Iy + J = 0$
Many websites say that "through appropriate change in variables," we can write the curve in the following simplified form:
$y^2 = x^3 + ax +b$
First of all I'm curious what that change in variables might look like.
Second of all, I'm wondering why exactly that's useful.
Say we have an elliptic curve in its general form, then we let, say, $y_1 = f(x,y)$ for the change in variables, and finally we have:
$Ax^3 + Bx^2 y + Cxy^2 + Dy^3 + Ex^2 + Fxy + Gy^2 + Hx + Iy + J = -y_1^2 + x^3 + ax +b = 0$,
what exactly have we accomplished? If we consider $y_1$ an indeterminate, we have a new curve.
Is it that we can start proving results for $-y_1^2 + x^3 + ax +b = 0$, and then for each point $(x, y_1)$ we have on this curve, we can translate it into a point $(x, f^{-1} (x, y_1))$ on the original curve?