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

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