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number > number theory > algebraic number theory > elliptic curve

geometry > algebraic geometry > algebraic variety > algebraic curve > elliptic curve

Terme préférentiel

elliptic curve

Définition(s)

In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point $O$ . An elliptic curve is defined over a field $K$ and describes points in $K 2$ , the Cartesian product of $K$ with itself. If the field's characteristic is different from 2 and 3, then the curve can be described as a plane algebraic curve which consists of solutions $(x, y)$ for:

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

for some coefficients $a$ and $b$ in $K$ . The curve is required to be non-singular, which means that the curve has no cusps or self-intersections. (This is equivalent to the condition $4 a 3 + 27 b 2 \neq 0$ , that is, being square-free in $x$ .) It is always understood that the curve is really sitting in the projective plane, with the point $O$ being the unique point at infinity. Many sources define an elliptic curve to be simply a curve given by an equation of this form.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Elliptic_curve)

Concept(s) générique(s)

Concept(s) spécifique(s)

Concept(s) associé(s)

Traductions

courbe elliptique

français

URI

http://data.loterre.fr/ark:/67375/PSR-JXHC9RBH-S

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