Elliptic curve

General equation

An elliptic curve is the set of solutions to a general equation in the third degree in two variables.

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

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

${}+rz+sw$

${}+t\qquad =0.$

The solutions to this equation are envisioned as points forming a curve in the Cartesian plane $(z,w)$. The general equation as given here has ten coefficients, g, h, j, k, m, p, q, r, s, and t.

Weierstraß normal form

Cf. Montgomery normal form.

Despite the apparent horrific complexity of the general equation, it has been found that its ten coefficients can be reduced to four, with only two of them left free, by a linear transformation of coördinates from the $(z,w)$ plane to another plane, labeled $(x,y)$.

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

For example, if $a=-1$ and $b=0$, then the elliptic curve $y^{2}=x^{3}-x$ may be plotted roughly as shown.

In Weierstraß normal form, the elliptic curve is symmetric (mirrored) about the x-axis, and the equation has no solutions in the real numbers where $x^{3}+ax+b<0$.