Elliptic curve

{wiki}

An elliptic curve is defined by numbers $a$ and $b$. The curve is the set of all points $(x, y)$ of the <real plane> that satisfy the <equation Definition of the elliptic curves>{full}

$$
y^2 = x^3 + ax + b
$$
{title=Definition of the <elliptic curves>}

\Image[https://upload.wikimedia.org/wikipedia/commons/thumb/d/db/EllipticCurveCatalog.svg/795px-EllipticCurveCatalog.svg.png]
{title=Plots of real elliptic curves for various values of $a$ and $b$}
{height=800}

<equation Definition of the elliptic curves> definies <elliptic curves> over any <field (mathematics)>, it doesn't have to the <real numbers>. Notably, the definition also works for <finite fields>, leading to <elliptic curve over a finite fields>, which are the ones used in <Elliptic-curve Diffie-Hellman> cyprotgraphy.
