Elliptic curve
An elliptic curve is a curve of degree 3 that can be described using the following formula:
[math]\displaystyle{ y^2=x^3+ax+b }[/math],
where [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math] are integers.
Elliptic Curve Media
A catalog of elliptic curves. The region shown is x, y ∈ [−3,3].(For (a, b) = (0, 0) the function is not smooth and therefore not an elliptic curve.)
Graphs of curves y2 = x3 − x and y2 = x3 − x + 1
example elliptic curves
Set of affine points of elliptic curve y2 = x3 − x over finite field F61.
Set of affine points of elliptic curve y2 = x3 − x over finite field F89.
Set of affine points of elliptic curve y2 = x3 − x over finite field F71.
An elliptic curve over the complex numbers is obtained as a quotient of the complex plane by a lattice Λ, here spanned by two fundamental periods ω1 and ω2. The four-torsion is also shown, corresponding to the lattice 14Λ containing Λ.
