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