2.12. Exercises 73 (1) Prove that the x-coordinate of 2P is given by x(2P ) = x4 0 2Ax2 0 8Bx0 + A2 4y02 . (2) Find a formula for y(2P ) in terms of x0 and y0. Exercise 2.12.17. The curve E/Q : y2 = x3 1572x has a rational point Q with x-coordinate x = x(Q) given by x = 224403517704336969924557513090674863160948472041 17824664537857719176051070357934327140032961660 2 . Show that there exists a point P E(Q) such that 2P = Q. Find the coordinates of P . (Hint: use PARI or Sage and Exercise 2.12.16.) Exercise 2.12.18. Let E : y2 = (x−e1)(x−e2)(x−e3) with ei Q, distinct, and such that e1 + e2 + e3 = 0. Additionally, suppose that e1 e2 = n2 and e2 e3 = m2 are squares. This exercise shows that, under these assumptions, there is a point P = (x0,y0) such that 2P = (e1,0), i.e., P is a point of exact order 4. (1) Show that e1 = n2+m2 3 , e2 = m2−2n2 3 , e3 = n2−2m2 3 . (2) Find A and B, in terms of n and m, such that x3+Ax+B = (x−e1)(x−e2)(x−e3). (Hint: Sage or PARI can be of great help here.) (3) Let p(x) = x4 −2Ax2 −8Bx+A2 −4(x3 +Ax+B)e1. Show that p(x0) = 0 if and only if x(2P ) = e1, and therefore 2P = (e1,0). (Hint: use Exercise 2.12.16.) (4) Express all the coefficients of p(x) in terms of n and m. (Hint: use Sage or PARI.) (5) Factor p(x) for (n, m) = (3,6), (3,12), (9,12),.... (6) Guess that p(x) = (x−a)2(x−b)2 for some a and b. Express all the coefficients of p(x) in terms of a and b. (7) Finally, compare the coefficients of p(x) in terms of a, b and n, m and find the roots of p(x) in terms of n, m. (Hint: compare first the coefficient of x3 and then the coefficient of x2.) (8) Write P = (x0,y0) in terms of n and m.
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