An algebraic curve over a field K is an equation , where f(X,Y) is a polynomial in X and Y with coefficients in K. A nonsingular algebraic curve is an algebraic curve over K which has no singular points over K. A point on an algebraic curve is simply a solution of the equation of the curve. A K-rational point is a point (X, Y) on the curve, where X and Y are in the field K.