Editing Elliptic curve (section)

==Elliptic curves over the real numbers==
[[Image:ECClines-3.svg|frame|right|Graphs of curves {{math|1=''y''<sup>2</sup> = ''x''<sup>3</sup> − ''x''}} and {{math|1=''y''<sup>2</sup> = ''x''<sup>3</sup> − ''x'' + 1}}]]
Although the formal definition of an elliptic curve requires some background in [[algebraic geometry]], it is possible to describe some features of elliptic curves over the [[real number]]s using only introductory [[algebra]] and [[geometry]].

In this context, an elliptic curve is a [[plane curve]] defined by an equation of the form

:<math>y^2 = x^3 + ax + b</math>

after a linear change of variables ({{mvar|a}} and {{mvar|b}} are real numbers). This type of equation is called a Weierstrass equation, and said to be in Weierstrass form, or Weierstrass normal form.

The definition of elliptic curve also requires that the curve be [[Singular point of an algebraic variety|non-singular]]. Geometrically, this means that the graph has no [[cusp (singularity)|cusps]], self-intersections, or [[Isolated point|isolated points]]. Algebraically, this holds if and only if the [[discriminant]], <math>\Delta</math>, is not equal to zero.

: <math>\Delta = -16\left(4a^3 + 27b^2\right) \neq 0</math>

The discriminant is zero when <math>a=-3k^2, b=2k^3</math>.

(Although the factor −16 is irrelevant to whether or not the curve is non-singular, this definition of the discriminant is useful in a more advanced study of elliptic curves.)<ref>{{Harvard citations|author=Silverman|year=1986|nb=yes|loc=III.1 Weierstrass Equations (p.45)}}</ref>

The real graph of a non-singular curve has ''two'' components if its discriminant is positive, and ''one'' component if it is negative. For example, in the graphs shown in figure to the right, the discriminant in the first case is 64, and in the second case is −368. Following the convention at [[Conic section#Discriminant]], ''elliptic'' curves require that the discriminant is negative.