Revision history [back]

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.

I tried to use the following codes in Sage.

R.<c1,c2> = QQ[]; Jacobian(11664c1^3c2^2 + 536544c1^3c2 + 6170256c1^3 + 67068c1^2c2^2 + 1542564c1^2c2 + 3085128c1c2^2 - 32393844c1c2 + 3085128c1 + 17739486c2^2 + 6941538c2)

But there is an error: NoEmbeddingError: not a sub-polytope of a reflexive polygon. How to find the normal form of the curve using Sage? Thank you very much.