Solve Cubic Equations: Elliptic Curve in Weierstrass Normal Form

[Newtime]

Is there an algorithmic way to put a given elliptic curve into Weierstrass normal form? If not, what's the general procedure?

 

[micromass]

Is there an algorithmic way to put a given elliptic curve into Weierstrass normal form? If not, what's the general procedure?

Yes. Say that C is a cubic. We start by picking a rational point O on the cubic. Then we take the tangent line of C at the rational point O. This will intersect the cubic in another rational point. We take the X-axis to be the tangent line at that other rational point. And we let the Y-axis to be any line through O.

By changing coordinates, you get an equation of the form

$xy^2+(ax+b)y=cx^2+d+e$

Multiply by x and change into the variable u=xy. This will get you something of the form

$$u^2+(ax+b)u=cx^3+dx^2+ex$$

Change the variable again by setting $v^2=u^2+(ax+b)$.

Now change the variables once more to obtain that c=1.