# Definition:Elliptic Curve

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## Definition

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:

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

where:

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

## Examples

### 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.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**elliptic curve** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**elliptic curve**