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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