Weierstraß normal form: Difference between revisions
Eliminating the skew term |
better explanation / def |
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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>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. | |||
== | == 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> | ||
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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 = 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> | ||
Latest revision as of 22:50, 8 February 2025
General equation
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle gz^3 + hz^2w + jzw^2 + kw^3}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {} + mz^2 + pzw + qw^2}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {} + rz + sw}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {} + t \qquad = 0.}
Linear transformation
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{bmatrix} z \\ w \end{bmatrix} = \begin{bmatrix} \alpha & \beta \\ \gamma & \delta \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z = \alpha x + \beta y + \zeta}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w = \gamma x + \delta y + \eta}
Problem
Substitute Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha x + \beta y + \zeta} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma x + \delta y + \eta} 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]
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y^2 = x^3 + ax + b}
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 “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 y2, and at that stage it is easy enough to eliminate the term in x2, as well.
Example: eliminating the skew term
Sometimes the Weierstraß equation is given in the form [2]
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w^2 + zw = z^3 + rz + t}
with an extra skew term Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle zw} . Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = w + \frac12z} and complete the square.
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y^2 - \frac14z^2 = z^3 + rz + t}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y^2 = z^3 + \frac14z^2 + rz + t}
Now let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=z+\frac1{12}} and complete the cube, viz.
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^3 = z^3 + \frac14z^2 + \frac1{48}z + \frac1{1728}}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y^2 = x^3 + \left(r-\frac1{48}\right)z + t-\frac1{1728}}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y^2 = x^3 + \left(r-\frac1{48}\right)\left(x-\frac1{12}\right) + t-\frac1{1728}}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y^2 = x^3 + \left[r-\frac1{48}\right]x + \left[t-\frac1{12}r +\frac1{864}\right]}
This is the normal Weierstraß form with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a=r-\frac1{48}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=t-\frac1{12}r +\frac1{864}} , and obviously rational points are mapped to rational points with the rational linear transformation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = w + \frac12z} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=z+\frac1{12}} .
- ↑ Arnold Kas. “Weierstrass Normal Forms and Invariants of Elliptic Surfaces.” Transactions of the American Mathematical Society, vol. 225, Jan 1977, pp. 259-266. PDF
- ↑ LMFDB Elliptic curves over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb Q} . https://www.lmfdb.org/EllipticCurve/Q/
