Elliptic Curves over Q - Elliptic Curves: Number Theory and Cryptography - page 205

Since y 2 = x 3 + Ax + B , we see that y 2 divides both terms on the left.

Therefore, y 2 4 A 3 +27 B 2 . |

COROLLARY 8.8
Let E be an elliptic curve over Q .Then he torsion subgroup of E ( Q ) is
finite.

PROOF A suitable change of variables puts the equation for E into Weier-
strass form with integer coecients. Theorem 8.7 now shows that there are
only finitely many possibilities for the torsion points.

Example 8.1
Let E be given by y 2 = x 3 +4. Then 4 A 3 +27 B 2 = 432. Let P =( x, y )bea
point of finite order in E ( Q ). Since 0 = x 3 + 4 has no rational solutions, we
have y = 0. Therefore, y 2 | 432, so

y = ± 1 , ± 2 , ± 3 , ± 4 , ± 6 , ± 12 .

Only y = ± 2 yields a rational value of x , so the only possible torsion points are
(0 , 2) and (0 , − 2). A quick calculation shows that 3(0 , ± 2) = ∞ . Therefore,
the torsion subgroup of E ( Q ) is cyclic of order 3.

Example 8.2
Let E be given by y 2 = x 3 +8. Then 4 A 3 +27 B 2 = 1728. If y =0,then
x = − 2. The point ( − 2 , 0) has order 2. If y =0,then y 2 | 1728, which means

that y| 24. Trying the various possibilities, we find the points (1 , ± 3) and
(2 , ± 4). However,

2(1 , 3) = ( − 7 / 4 , − 13 / 8) and 2(2 , 4) = ( − 7 / 4 , 13 / 8) .

Since these points do not have integer coordinates, they cannot have finite
order. Therefore, (1 , 3) and (2 , 4) cannot have finite order. It follows that the
torsion subgroup of E ( Q )is {∞, ( − 2 , 0) } .( Remark: Thefactthat2(1 , 3) =
− 2(2 , 4) leads us to suspect, and easily verify, that (1 , 3) + (2 , 4) = ( − 2 , 0).)

Suppose we use the Lutz-Nagell theorem and obtain a possible torsion point
P . How do we decide whether or not it's a torsion point? In the previous
example, we multiplied P by an integer and obtained a nontorsion point.
Therefore, P was not a torsion point. In general, the Lutz-Nagell theorem
explicitly gives a finite list of possibilities for torsion points. If P is a torsion
point, then, for every n , the point nP must either be ∞ or be on that list.
Since there are only finitely many points on the list, either we'll have nP = mP
for some m = n , in which case P is torsion and ( n − m ) P = ∞ ,orsome