2.12. Exercises 73
(1) Prove that the x-coordinate of 2P is given by
x(2P ) =
− 8Bx0 +
(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
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 =
, e2 =
, e3 =
(2) Find A and B, in terms of n and m, such that
(x−e1)(x−e2)(x−e3). (Hint: Sage or PARI can be of great
(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) =
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
(8) Write P = (x0,y0) in terms of n and m.