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28 1. Elementary Properties of the Chord-Tangent Group Law on a Cubic Curve

2
3
2
3
few multiples of P:9P = (−20/7 , −435/7 ),16P = (a/65 , b/65 ) for some
2
2

integers a and b, and 51P = (a /N , b /N ), where the natural number N has 32

digits and a , b are integers.
Exercises
2
3
1. Find 7P, −7P,8P, −8P,9P, −9P,10P,and −10P on y + y = x − x in (1.6).
Using Mazur’s theorem in §5 of the Introduction, argue that P has infinite order where
P = (0, 0).
2
3
2. Show that P = (0, 2) is a point of order 3 on y = x + 4.
3
2
3. Show that P = (2, 4) is a point of order 4 on y = x + 4x.
2 3
4. Show that P = (2, 3) is a point of order 6 on y = x + 1.
5. Show that P = (−12, 108) is a point of order 5 on
2 3
y = x − 16 · 27x + 19 · 16 · 27.
6. Let E denote the elliptic curve defined by the cubic equation
2
y = x(x − 1)(x + 9).
Find a subgroup of order four in E(Q), show that (−1, 4) = P is on E(Q) and not in this
subgroup, and calculate nP for n between −7and +7. Using Mazur’s theorem in §5of
the Introduction, argue that P has infinte order.
Remark. The exercises will also illustrate results in Chapter 5.
§2. Illustrations of the Elliptic Curve Group Law
Before discussing some more examples, we make some remarks about special forms
of the normal form in which certain coefficients are zero. If 2 = 0 in the field k, i.e.,
the characteristic of k is different from 2, then in the normal form
3
2
2
y + y(a 1 x + a 3 ) = x + a 2 x + a 4 x + a 6 ,
we can complete the square on the left-hand side
(a 1 x + a 3 ) 2 2
2 a 1 x + a 3
y + y(a 1 x + a 3 ) + = y + .
4 2
With a change of variable y to y − (a 1 x + a 3 )/2, we obtain the equivalent cubic
2
equation y = f (x), where f (x) is a cubic polynomial in x.
2
(2.1) Remark. If the equation for E is y = f (x), where f (x) is a cubic polyno-
mial, then the negative of an element is given by −(x, y) = (x, −y). Furthermore,
the cubic will be nonsingular if and only if f (x) has no repeated roots.

The reason that we might consider normal forms with terms a 1 xy and a 3 y is that
the cubic might have a particularly simple form as in the case (1.3). These terms are
always necessary potentially for a theory in characteristic 2, e.g., over the field F 2 of
two elements.