Weierstraß normal form

General equation

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

$gz^{3}+hz^{2}w+jzw^{2}+kw^{3}$

${}+mz^{2}+pzw+qw^{2}$

${}+rz+sw$

${}+t\qquad =0.$

Linear transformation

${\begin{bmatrix}z\\w\end{bmatrix}}={\begin{bmatrix}\alpha &\beta \\\gamma &\delta \end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}$

$z=\alpha x+\beta y+\zeta$

$w=\gamma x+\delta y+\eta$

Problem

Substitute $\alpha x+\beta y+\zeta$ and $\gamma x+\delta y+\eta$ 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

$y^{2}=x^{3}+ax+b$

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

$w^{2}+zw=z^{3}+rz+t$

with an extra skew term $zw$. Let $y=w+{\frac {1}{2}}z$ and complete the square.

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

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

Now let $x=z+{\frac {1}{12}}$ and complete the cube, viz.

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

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

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

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

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 $y=w+{\frac {1}{2}}z$ and $x=z+{\frac {1}{12}}$.