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Graphs of curves y 2 = x 3 − x and y 2 = x 3 − 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 numbers using only introductory algebra and geometry.
They are named elliptic functions because they come from elliptic integrals. Those integrals are in turn named elliptic because they first were encountered for the calculation of the arc length of an ellipse. Important elliptic functions are Jacobi elliptic functions and the Weierstrass ℘-function.
A ℘-function together with its derivative can be used to parameterize elliptic curves and they generate the field of elliptic functions with respect to a given period lattice. Symbol for Weierstrass ℘ {\displaystyle \wp } -function
Every elliptic curve E over C is a complex torus, and thus can be identified with a rank 2 lattice; that is, a two-dimensional lattice of C. This lattice can be rotated and scaled (operations that preserve the isomorphism class), so that it is generated by 1 and τ ∈ H. This lattice corresponds to the elliptic curve = () (see Weierstrass ...
Plot of the degenerate Jacobi curve (x 2 + y 2 /b 2 = 1, b = ∞) and the twelve Jacobi Elliptic functions pq(u,1) for a particular value of angle φ. The solid curve is the degenerate ellipse ( x 2 = 1) with m = 1 and u = F ( φ ,1) where F (⋅,⋅) is the elliptic integral of the first kind.
Elliptic curve; Watt's curve; Curves with genus > 1. Bolza surface (genus 2) Klein quartic (genus 3) ... Cardiac function curve; Dose–response curve; Growth curve ...
In mathematics, the rank of an elliptic curve is the rational Mordell–Weil rank of an elliptic curve defined over the field of rational numbers or more generally a number field K. Mordell's theorem (generalized to arbitrary number fields by André Weil ) says the group of rational points on an elliptic curve has a finite basis .
Pages in category "Elliptic curves" The following 52 pages are in this category, out of 52 total. ... Hasse's theorem on elliptic curves; Heegner point; Height function;