Let $C$ be the following curve in $\mathbb{C}^2$.
\begin{align}
& 11664\, {c\_1}^3\, {c\_2}^2 + 536544\, {c\_1}^3\, c\_2 + 6170256\, {c\_1}^3 + 67068\, {c\_1}^2\, {c\_2}^2 + 1542564\, {c\_1}^2\, c\_2 \\
& + 3085128\, c\_1\, {c\_2}^2 - 32393844\, c\_1\, c\_2 + 3085128\, c\_1 + 17739486\, {c\_2}^2 + 6941538\, c\_2 = 0.
\end{align}
I checked that this curve has genus $1$ using Sage. Therefore it is an elliptic curve. How to change coordinates such that the equation of this curve is of the form $y^2 = f(x)$, where $f$ is some polynomial. Thank you very much. 



How to find the normal form of an elliptic curve?
First, you need a nonsingular point of the curve. The only singular points are the two points at infinity, so the point $(0,0)$ is safe. 


Now, you can use magma to, first, define the affine curve, then its projective closure $CP$, and then call the function EllipticCurve(CP, P);
where $P$ is the point $CP![0,0,1]$. It gives the curve in generalized Weierstrass form, and the morphism. Then you can compute other Weierstrass models. The minimal model over $\mathbb Q$ is 
$$y^2 = x^3 - x^2 + 1018049744x - 672911244106688. $$


