Weierstraß normal form

General equation

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

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

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.}

Eliminating the skew term

Sometimes the Weierstraß equation is given in the form

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.

$x^{3}=z^{3}+{\frac {1}{4}}z^{2}+{\frac {1}{48}}z+{\frac {1}{1728}}$

$y^{2}=z^{3}+{\frac {1}{4}}z^{2}+rz+t$

$y^{2}=x^{3}+\left(r-{\frac {1}{48}}\right)z+t-{\frac {1}{1728}}$

$y^{2}=x^{3}+\left(r-{\frac {1}{48}}\right)\left(x-{\frac {1}{12}}\right)+t-{\frac {1}{1728}}$

$y^{2}=x^{3}+\left[r-{\frac {1}{48}}\right]x+\left[t-{\frac {1}{12}}r+{\frac {1}{864}}\right]$

This is the normal Weierstraß form with $a=r-{\frac {1}{48}}$ and $b=t-{\frac {1}{12}}r+{\frac {1}{864}}$, 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}} .