Normal form for Cubic

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

    micromass 18,532
    Staff Emeritus
    Science Advisor

    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.
     
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