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§1. Chord-Tangent Computational Methods on a Normal Cubic Curve 27

0 = 2(−1, −1) = 2 · 3P = 6P since the tangent at (−1, −1) is vertical, and we
deduce that P is a point of order 6. In particular
4P =−2P =−(0, 0) = (0, −1)

and

5P =−P =−(1, 1) = (1, −1).

Thus P m = mP,and O together with the five points P,2P,3P,4P, and 5P shown
on the cubic E form a cyclic subgroup of order 6 in E(Q) in (1.3).

If we study the question of when the product of three consecutive numbers y(y +
3
1) is the product of three consecutive numbers (x − 1)x(x + 1) = x − x, we are led
to the following example of an elliptic curve.
2
(1.6) Example. The elliptic curve E defined by the normal cubic equation y + y =
3
x − x has six obvious points on it (0, 0), (1, 0), (−1, 0), (0, −1), (1, −1),and
(−1, −1).If P = (0, 0), then these points are all in the subgroup generated by P as
with P = (1, 1) in (1.5), but in this case P generates an infinite cyclic group.

In general odd multiples of P are on the closed component of the curve which
contains P = (0, 0) and the even multiples of P are on the open component which
is closed up to a circle by adding O at infinity. We have the following values for a