Elliptic Curves over Q - Elliptic Curves: Number Theory and Cryptography - page 224

We claim that

E ( Q ) Z 2 ⊕ Z 2 ⊕ Z .

We know that the rank r is at least 1, because there is a point ( − 4 , 6) of
infinite order. The problem is to show that the rank is exactly 1.
Consider the map

φ : E ( Q ) → ( Q × / Q × 2 ) ⊕ ( Q × / Q × 2 ) ⊕ ( Q × / Q × 2 )

of Theorem 8.14 defined by

( x, y ) → ( x, x − 5 ,x +5)

when y = 0. Therefore,

φ ( − 4 , 6) = ( − 1 , − 1 , 1) ,

wherewehaveusedthefactthat − 4and − 9areequivalentto − 1 mod squares.

Also, from Theorem 8.14,

φ ( ∞ )=(1 , 1 , 1)
φ (0 , 0) = ( − 1 , − 5 , 5)
φ (5 , 0) = (5 , 2 , 10)
φ (

− 5 , 0) = ( − 5 , − 10 , 2) .

Since φ is a homomorphism, we immediately find that φ ( − 4 , 6) times any of
these triples is in the image of φ ,so

(1 , 5 , 5) , ( − 5 , − 2 , 10) , (5 , 10 , 2)

correspond to points.
If we write

x = au 2
x − 5= bv 2
x +5= cw 2 ,

we have φ ( x, y )=( a, b, c ). From Proposition 8.13, we may assume

a, b, c ∈{± 1 , ± 2 , ± 5 , ± 10 } .

Also, abc is a square, so c is determined by a, b . Therefore, we'll often ignore
c and concentrate on the possibilities for a, b . There are 64 possible pairs a, b .
So far, we have 8 pairs that correspond to points. Let's record them in a list,
which we'll refer to as L in the following:

L = { (1 , 1) , (1 , 5) , ( − 1 , − 1) , ( − 1 , − 5) , (5 , 2) , (5 , 10) , ( − 5 , − 2) , ( − 5 , − 10) }.