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§2. Illustrations of the Elliptic Curve Group Law 33

the group E(Q) is the cyclic group of four elements

1 1 1

0, 1, , , 0 , 1, − ,
2 2 2
for if E(Q) contained other rational points (x, y) there would be corresponding
4
4
4
(u,v,w) solutions to the Fermat equation u + v = w , see Exercise 2. This equa-
tion for the curve E takes a simplier form under the substitutions x/4 for x − 1/2
and y/8 for y. The new equations are

y 2 x x 1
2
= +
8 4 16 4
or
2 3
y = x + 4x.
This is the curve considered in Exercise 3 to §1 where the question was to show that a
certain point was of order 4; in fact, from the relation to the Fermat curve we deduce
3
2
that E(Q) is cyclic of order 4 for the curve defined by y = x + 4x.
Exercises
2
2
3
1. Find +6P, −6P, +7P, −7P,8P,and −8P on y + y = x +x in (2.4). Using Mazur’s
theorem in §5 of the Introduction, argue that P has infinite order.
2. In (2.5) and in (2.6) write u and v as functions of (w, x, y).
3
3
3. For u + v = c consider the change of variable
12c y
u + v = and u − v = .
x 3x
Show that
12c u − v
x = and y = 36c ,
u + v u + v
and that the curve is transformed into the cubic in normal form
2 3 2
y = x − 432c .
Determine the group of rational points on this elliptic curve for c = 1, for c = 2.
4. Show that P = (3, 12) is a point of order 8 on
2 3 2
y = x − 14x + 81x.
5. Show that P = (1, 0) is a point of order 7 on
2 3 2
y + xy + y = x − x − 3x + 3.
6. Calculate all multiples nP of P = (3, 8) on
2 3
y = x − 43x + 166.
Find the order of (3, 8) and of (11, 32) on this curve.