I came across the following change of variables to turn equation into an elliptic curve.
(Source: https://www.youtube.com/watch?v=6eZQu120A80&t=3022s&ab_channel=ImperialCollegeLondon, the part I'm talking about starts at 23:20)
We are given: $\frac{(t^2+1)^2}{c^2}=t-t^3$, where $t \in \Bbb R$ and $c$ is a positive constant.
Now if you define $Y = \frac{t^2+1}{c}$ and $X = -t$, our equation becomes $Y^2 = X^3 - X$ which is an elliptic curve.
In the video, he says studying this elliptic curve can give solutions about the original equation.
If this transformation is any useful, how does it preserve properties of the original equation? That is: $\frac{(t^2+1)^2}{c^2}=t-t^3$
We defined $Y = \frac{t^2+1}{c}$, which sets the range of Y as ($\infty \gt Y\ge 1/c$) and X as ($\infty \ge X \ge -\infty$) but if we preserve this range, our elliptic curve is not an elliptic curve? So how can you use the properties of elliptic curve and get something useful out of the original expression?
More general question would be, when is it useful to turn one equation to another known equation (with good properties) by change of variable? Are there any requirements for it to preserve properties?