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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