Quadratic twist of an elliptic curve

Question

I found this page: http://en.wikipedia.org/wiki/Twists_of_curves#Quadratic_twist

which tells me $dy^2=x^3+a_2x^2+a_4x+a_6$ is equivalent to $y^2=x^3+da_2x^2+d^2a_4x+d^3a_6$. Why is this equivalent (for $d$ as given on that page)?

Answer 1

Put $y = y'/d^2$ and $x=x'/d$. Then

$$d^{-3}y'^2 = d^{-3}x'^3 + a_2d^{-2}x'^2 + a_4d^{-1}x' + a_6.$$

Multiply both sides by $d^3$.