Rational Point

A K-rational point is a point (X,Y) on an algebraic curve f(X,Y)=0, where X and Y are in a field K. For example, rational point in the field Q of ordinary rational numbers is a point (X,Y) satisfying the given equation such that both X and Y are rational numbers.

The rational point may also be a point at infinity. For example, take the elliptic curve


and homogenize it by introducing a third variable Z so that each term has degree 3 as follows:


Now, find the points at infinity by setting Z=0, obtaining


Solving gives X=0, Y equal to any value, and (by definition) Z=0. Despite freedom in the choice of Y, there is only a single point at infinity because the two triples (X_1, Y_1, Z_1), (X_2, Y_2, Z_2) are considered to be equivalent (or identified) only if one is a scalar multiple of the other. Here, (0, 0, 0) is not considered to be a valid point. The triples (a, b, 1) correspond to the ordinary points (a, b), and the triples (a, b, 0) correspond to the points at infinity, usually called the line at infinity.

The rational points on elliptic curves over the finite field GF(q) are 5, 7, 9, 10, 13, 14, 16, ... (OEIS A005523).

See also

Elliptic Curve, Line at Infinity, Point at Infinity

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Sloane, N. J. A. Sequence A005523/M3757 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Rational Point

Cite this as:

Weisstein, Eric W. "Rational Point." From MathWorld--A Wolfram Web Resource.

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