Weierstraß normal form - Revision history

Rational Point: better explanation / def

2025-02-08T22:50:05Z

better explanation / def

← Older revision		Revision as of 22:50, 8 February 2025
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	::::<math>{} + t \qquad = 0.</math>		::::<math>{} + t \qquad = 0.</math>


	== Linear transformation ==		== Linear transformation ==
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	Substitute <math>\alpha x + \beta y + \zeta</math> and <math>\gamma x + \delta y + \eta</math> 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<ref>Arnold Kas. “Weierstrass Normal Forms and Invariants of Elliptic Surfaces.” ''Transactions of the American Mathematical Society,'' vol. 225, Jan 1977, pp. 259-266. [https://www.ams.org/journals/tran/1977-225-00/S0002-9947-1977-0422285-X/S0002-9947-1977-0422285-X.pdf PDF]</ref>		Substitute <math>\alpha x + \beta y + \zeta</math> and <math>\gamma x + \delta y + \eta</math> 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<ref>Arnold Kas. “Weierstrass Normal Forms and Invariants of Elliptic Surfaces.” ''Transactions of the American Mathematical Society,'' vol. 225, Jan 1977, pp. 259-266. [https://www.ams.org/journals/tran/1977-225-00/S0002-9947-1977-0422285-X/S0002-9947-1977-0422285-X.pdf PDF]</ref>

	:<math>y^2 = x^3 + ax + b.</math>		:<math>y^2 = x^3 + ax + b</math>

			This is the [[Weierstraß normal form]] of an elliptic curve.
			Sometimes the general equation is given in partially simplified form as, e.g. LMFDB’s [https://www.lmfdb.org/knowledge/show/ec.weierstrass_coeffs “Weierstrass equation or model”] where some of the coefficients have already been eliminated. A minimal model for an elliptic curve often retains some coefficients which would be eliminated by a transformation to a strict Weierstraß normal form, when the goal of simplifying the equations is to avoid coefficients of excessive [[height]]. However, the Weierstraß normal form eliminates all skew terms and all terms in ''y'' except for ''y''<sup>2</sup>, and at that stage it is easy enough to eliminate the term in ''x''<sup>2</sup>, as well.

	== ~~Eliminating~~ the skew term ==		== Example: 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>		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>

Rational Point: dup formula

2025-01-03T12:47:11Z

dup formula

← Older revision		Revision as of 12:47, 3 January 2025
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	:<math>x^3 = z^3 + \frac14z^2 + \frac1{48}z + \frac1{1728}</math>		:<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)z + t-\frac1{1728}</math>

Rational Point: Eliminating the skew term

2025-01-03T12:44:45Z

Eliminating the skew term

← Older revision		Revision as of 12:44, 3 January 2025
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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>.

Rational Point: new category + ref

2024-12-19T22:30:13Z

new category + ref

← Older revision		Revision as of 22:30, 19 December 2024
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			[[Category:Normal forms]]

	== General equation ==		== General equation ==
	[[Image:Karl-weierstrass-1-39ba31-640.jpg\|right\|frame\|Karl		[[Image:Karl-weierstrass-1-39ba31-640.jpg\|right\|frame\|Karl
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	== Problem ==		== Problem ==

	Substitute <math>\alpha x + \beta y + \zeta</math> and <math>\gamma x + \delta y + \eta</math> 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		Substitute <math>\alpha x + \beta y + \zeta</math> and <math>\gamma x + \delta y + \eta</math> 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<ref>Arnold Kas. “Weierstrass Normal Forms and Invariants of Elliptic Surfaces.” ''Transactions of the American Mathematical Society,'' vol. 225, Jan 1977, pp. 259-266. [https://www.ams.org/journals/tran/1977-225-00/S0002-9947-1977-0422285-X/S0002-9947-1977-0422285-X.pdf PDF]</ref>

	:<math>y^2 = x^3 + ax + b.</math>		:<math>y^2 = x^3 + ax + b.</math>

Rational Point: Weierstraß normal form - homework problem ??

2024-12-17T22:41:13Z

Weierstraß normal form - homework problem ??

New page

== General equation ==
[[Image:Karl-weierstrass-1-39ba31-640.jpg|right|frame|Karl
Theodor Wilhelm Weierstraß (31 Oct 1815 – 19 Feb 1897)]]

:<math>gz^3 + hz^2w + jzw^2 + kw^3</math>

::<math>{} + mz^2 + pzw + qw^2</math>

:::<math>{} + rz + sw</math>

::::<math>{} + t \qquad = 0.</math>

== Linear transformation ==

<math>\begin{bmatrix} z \\ w \end{bmatrix}
= \begin{bmatrix} \alpha & \beta \\
\gamma & \delta \end{bmatrix}
\begin{bmatrix} x \\ y \end{bmatrix}
</math>

<math>z = \alpha x + \beta y + \zeta</math>

<math>w = \gamma x + \delta y + \eta</math>

== Problem ==

Substitute <math>\alpha x + \beta y + \zeta</math> and <math>\gamma x + \delta y + \eta</math> 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

:<math>y^2 = x^3 + ax + b.</math>