2.4. The group structure on E(Q) 27

Figure 2. The rational points on

y2

=

x3

+ 1, except the

point at ∞.

• The point 4P can be found by adding 3P and P . The third

point of intersection of E and the line through P and 3P is

R = 2P = (0,1), and so 4P = P + 3P = (0, −1).

• We find 5P by adding 4P and P . Notice that the line that

goes through 4P = (0, −1) and P = (2,3) is tangent at

(2,3), so the third point of intersection is P . Thus, 5P =

4P + P = (2, −3).

• Finally, 6P = P + 5P but 5P = (2, −3) = −P . Hence,

6P = P + (−P) = O, the point at infinity.

This means that P is a point of finite order, and its order equals

6. See Figure 2 (the Sage code for this graph can be found in the

Appendix A.1.3).

The addition law can be defined more generally on any smooth

projective cubic curve E : f(X, Y, Z) = 0, with a given rational point

O. Let P, Q ∈ E(Q) and let L be the line that goes through P and