Weierstraß normal form: Difference between revisions

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Eliminating the skew term
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:<math>y^2 = x^3 + ax + b.</math>
:<math>y^2 = x^3 + ax + b.</math>
== Eliminating the skew term ==
Sometimes the Weierstraß equation is given in the form <ref>LMFDB Elliptic curves over <math>\mathbb Q</math>. https://www.lmfdb.org/EllipticCurve/Q/</ref>
:<math>w^2 + zw = z^3 + rz + t</math>
with an extra skew term <math>zw</math>. Let <math>y = w + \frac12z</math> and complete the square.
:<math>y^2 - \frac14z^2 = z^3 + rz + t</math>
:<math>y^2 = z^3 + \frac14z^2 + rz + t</math>
Now let <math>x=z+\frac1{12}</math> and complete the cube, viz.
:<math>x^3 = z^3 + \frac14z^2 + \frac1{48}z + \frac1{1728}</math>
:<math>y^2 = z^3 + \frac14z^2 + rz + t</math>
:<math>y^2 = x^3 + \left(r-\frac1{48}\right)z + t-\frac1{1728}</math>
:<math>y^2 = x^3 + \left(r-\frac1{48}\right)\left(x-\frac1{12}\right) + t-\frac1{1728}</math>
:<math>y^2 = x^3 + \left[r-\frac1{48}\right]x + \left[t-\frac1{12}r +\frac1{864}\right]</math>
This is the normal Weierstraß form with <math>a=r-\frac1{48}</math> and <math>b=t-\frac1{12}r +\frac1{864}</math>, and obviously rational points are mapped to rational points with the rational linear transformation <math>y = w + \frac12z</math> and <math>x=z+\frac1{12}</math>.

Revision as of 12:44, 3 January 2025


General equation

Karl Theodor Wilhelm Weierstraß (31 Oct 1815 – 19 Feb 1897)

Linear transformation

Problem

Substitute and for z and w in the general equation, simplify by collecting like terms in respective powers of x and y, and solve for α, β, γ, δ, ζ, η, a and b so that[1]


Eliminating the skew term

Sometimes the Weierstraß equation is given in the form [2]

with an extra skew term . Let and complete the square.

Now let and complete the cube, viz.

This is the normal Weierstraß form with and , and obviously rational points are mapped to rational points with the rational linear transformation and .

  1. Arnold Kas. “Weierstrass Normal Forms and Invariants of Elliptic Surfaces.” Transactions of the American Mathematical Society, vol. 225, Jan 1977, pp. 259-266. PDF
  2. LMFDB Elliptic curves over . https://www.lmfdb.org/EllipticCurve/Q/