Number Theory: An Introduction to Mathematics

554 XIII Connections with Number Theory

Singular cases

Node:d=0,a (^0) Cusp:d = a = 0
x
y y
P x P
Non-singular cases
y y
x x
d<0 d>0
a, b:y^2 =x^3 +ax + b (a,b∈ ;d = 4a^3 + 27b^2 )
Fig. 1.Cubic curves overR.
The different types of curve which arise whenK=Ris the field of real numbers
are illustrated in Figure 1. The unique point at infinity 0 may be thought of as being
at both ends of they-axis. (In the case of a node, Figure 1 illustrates the situation for
x 0 >0. Forx 0 <0 the singular point is an isolated point of the curve.)
Suppose now thatKis any field of characteristic= 2 ,3 and that the curveCa,bhas
zero discriminant. Because of the geometrical interpretation whenK=R, the unique
singular point of the curveCa,bis said to be anodeifa=0andacuspifa=0.
In the cusp case, if we putT=Y/X, then the cubic curve has the parametrization