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].
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