How to find the normal form of an elliptic curve?

Question

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.

Answer 1

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 $$y^2 = x^3 - x^2 + 1018049744x - 672911244106688. $$

Answer 2

You have a form $$ y^2(Ax^3+Bx^3+Cx^2+Dx+E)+y(Fx^3+Gx^2+Hx+I)+(Jx^3+Kx^2+Lx+M)=0$$

Some of the alphabet are zero in your case. You can complete the square in y.

$$y^2f(x)+yg(x)+h(x)=0\rightarrow [2f(x)y+g(x)]^2+4f(x)h(x)-g(x)^2=0$$

Start with analysing the simpler points of your multinomial where $g(x) = 0$ (reduce). This will inform the locus of transition points in the (xy) plane. After this you have at least three options.

The first is to calculate the curvature by transforming into parametric coordinates and locate the critical points.

The second is to regularise beyond the euclidean-normalisable region by extending $y$ to $y+re^{i\theta}$ to test stability and check for splitting.

The third is to examine the discriminant $g(x)^2-4f(x)h(x)$. It is at most degree six. If it is solvable then determine the group structure on permutations of the roots; this will inform the classification of affine structure.