How to see the elliptic curves $y^2=x^3+ax+b$ and $y^2=x^3+ac^{-4}x+bc^{-6}$ are isomorphic ?
Here $a,~b,~,c \in \mathbb{Q}$ and $c \neq 0$.
If I scale $x$ and $y$ by $(x,y,z) \mapsto (c^2x,c^3y,z)$ in the 2nd curve, we get $$ (c^3y)^2=(c^2x)^3+ac^{-4} \cdot c^2x+bc^{-6}.$$ Simplifying , we get $$c^6y^2=c^6x^3+ac^{-2}x+bc^{-6}.$$
Cancelling out by $c^6$ both sides, we get $$y^2=x^3+ac^{-8}+bc^{-12}.$$
So not isomorphic if my calculation is correct.
Edit: I got it from the below comment, the map works for opposite way to my above calculation