Solve Cubic Equations: Elliptic Curve in Weierstrass Normal Form

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The discussion confirms that there is an algorithmic method to convert a given elliptic curve into Weierstrass normal form. The process begins by selecting a rational point O on the cubic curve C and determining the tangent line at that point, which intersects the curve at another rational point. By applying a series of coordinate transformations, including setting the X-axis as the tangent line at the new point and the Y-axis as a line through O, the curve can be expressed in the form xy² + (ax + b)y = cx² + d + e. Further variable substitutions lead to the final form where c = 1.

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

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