2.9. Descent and the weak Mordell-Weil theorem 49

Are P , Q and R independent? In order to find out, we find the elliptic

height matrix associated to {P, Q, R}, using PARI or Sage:

H =

⎛

⎝

P, P Q, P R, P

P, Q Q, Q R, Q

P, R Q, R R, R

⎞

⎠

=

⎛

⎝

7.397 . . . −3.601 . . . 3.795 . . .

−3.601 . . . 6.263 . . . 2.661 . . .

3.795 . . . 2.661 . . . 6.457 . . .

⎞

⎠

.

The determinant of H seems to be very close to 0 (PARI returns

3.368 ·

10−27).

Hence Cor. 2.8.6 suggests that P , Q and R are not

independent. If we find the (approximate) kernel of H with PARI, we

discover that the (column) vector (1,1, −1) is approximately in the

kernel, and therefore, P + Q − R may be a torsion point. Indeed, the

point P +Q−R = (0,0) is a torsion point of order 2 on E(Q). Hence,

P , Q and R are linearly dependent over Z.

Instead, let P1 = (−1681,25543), P2 = (−338,26), a third point

P3 = (577/16,332929/64) and let H be the elliptic height matrix

associated to {Pi}i=1.

3

Then det(H ) = 101.87727 . . . is non-zero and,

therefore, {Pi}i=1

3

are linearly independent and the rank of E/Q is at

least 3.

2.9. Descent and the weak Mordell-Weil

theorem

In the previous sections we have seen methods to calculate the torsion

subgroup of an elliptic curve E/Q, and also methods to check if a

collection of points are independent modulo torsion. However, we

have not discussed any method to find points of infinite order. In this

section, we briefly explain the method of descent, which facilitates the

search for generators of the free part of E(Q). Unfortunately, the

method of descent is not always successful! We will try to measure

the failure of the method in the following section. The method of

descent (as explained here) is mostly due to Cassels. For a more

detailed treatment, see [Was08] or [Sil86]. A more general descent

algorithm was laid out by Birch and Swinnerton-Dyer in [BSD63].