2.12. Exercises 73

(1) Prove that the x-coordinate of 2P is given by

x(2P ) =

x0

4

− 2Ax0

2

− 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.