Normal form for Cubic


by Newtime
Tags: cubic, form, normal
Newtime
Newtime is offline
#1
Aug8-11, 05:32 PM
P: 353
Is there an algorithmic way to put a given elliptic curve into Weierstrass normal form? If not, what's the general procedure?
Phys.Org News Partner Science news on Phys.org
Cougars' diverse diet helped them survive the Pleistocene mass extinction
Cyber risks can cause disruption on scale of 2008 crisis, study says
Mantis shrimp stronger than airplanes
micromass
micromass is online now
#2
Aug8-11, 08:15 PM
Mentor
micromass's Avatar
P: 16,692
Quote Quote by Newtime View Post
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

[itex]xy^2+(ax+b)y=cx^2+d+e[/itex]

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

[tex]u^2+(ax+b)u=cx^3+dx^2+ex[/tex]

Change the variable again by setting [itex]v^2=u^2+(ax+b)[/itex].

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


Register to reply

Related Discussions
Another form of cubic folium equation Differential Geometry 2
How are cubic equations converted into this form? General Math 2
Game Theory: converting extensive form to normal form Set Theory, Logic, Probability, Statistics 1
Question about the General form to normal form of Diff Eq Calculus & Beyond Homework 1
Solving a cubic (or rather simplifying a real root in complex form) Calculus & Beyond Homework 2