Definition:Elliptic Curve

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An elliptic curve is a plane curve $C$ embedded in the Cartesian plane such that:

$C$ can be defined by a polynomial equation such that:
the coefficients of $C$ are rational
the genus of $C$ is $1$
$C$ has at least $1$ rational lattice point.

Standard Form

Let $C$ be an elliptic curve embedded in the Cartesian plane.

$C$ is expressed in standard form when the equation defining its locus is in the form:

$y^2 = x^3 + a x + b$


$a$ and $b$ are rational numbers
the cubic $x^3 + a x + b$ has distinct zeroes.


Arbitrary Example

The equation:

$y^2 = x^3 + 17$

describes an elliptic curve.

This has the rational lattice points $\tuple {-2, 3}$, $\tuple {2, 5}$, $\tuple {\dfrac 1 4, \dfrac {33} 8}$, $\tuple {-1, 4}$ and many more.

Also see

  • Results about elliptic curves can be found here.

Historical Note

The study of elliptic curves and their rational lattice points was instrumental in the proof of Fermat's Last Theorem.