TITLE: Turning an elliptic curve over C into a complex torus
QUESTION [10 upvotes]: I have been reading a lot about the Weierstrass $\wp$ function and I understand the parameterization of an elliptic curve with the elliptic function( i.e. $x=\wp(z)$ and $y=\wp^\prime(z)$). I would like to parameterize my own elliptic curve, but I'm running into some difficulty. For example, with the curve $y^2=x^3+2x^2−3x$, all I want to do is find $\wp(z)$ and $\wp^\prime(z)$ for some $z$ (should I use $\omega_1$ and $\omega_2$?) and I would like to compute $g_2$ and $g_3$. I guess I could have saved some space by just saying that I want to find a parameterization between $y^2=x^3+2x^2−3x$ and $y^2=4x^3−g_2x−g_3$. Is this trivial and I'm just missing something? I know Sage will handle finding at least the periods.

REPLY [3 votes]: Quite likely I am misunderstanding what you want to do, but if you have $y^2=x^3+2x^2-3x$, you can use the depression substitution $x=t-\frac23$ to yield the new cubic
$$y^2=t^3-\frac{13}{3}t+\frac{70}{27}$$
With the further substitution $t=\sqrt[3]{4}u$, we have
$$y^2=4u^3-\frac{13\sqrt[3]{4}}{3}u+\frac{70}{27}$$
which means we have the invariants $g_2=\frac{13\sqrt[3]{4}}{3}$ and $g_3=-\frac{70}{27}$ (with respect to the new coordinates).
As already noted, from these invariants, you can express the periods in terms of the complete elliptic integral of the first kind (see e.g. the DLMF for the needed formulae), which is most efficiently evaluated with the arithmetic-geometric mean iteration.

As Brent notes in the comments, some take the Weierstrass form to have a leading coefficient of $1$ instead of $4$; that is, the standard form looks like
$$y^2=x^3-g_2 x-g_3$$
In this case, one just performs the depression substitution, yielding $g_2=\dfrac{13}{3}$.