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
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