2.12. Exercises 71

(2) Let y2 = x(x + 5)(x + 10) and P = (−9,6). Find nP for

n = 1,...,12. Compare the x-coordinates of nP with the

list given at the end of Example 1.1.1, and write down the

next three numbers that belong in the list.

Exercise 2.12.6. Let E/Q be an elliptic curve given by a Weierstrass

equation of the form y2 = f(x), where f(x) ∈ Z[x] is a monic cubic

polynomial with distinct roots (over C).

(1) Show that P = (x, y) ∈ E is a torsion point of exact order

2 if and only if y = 0 and f(x) = 0.

(2) Let E(Q)[2] be the subgroup of E(Q) formed by those ra-

tional points P ∈ E(Q) such that 2P = O. Show that the

size of E(Q)[2] may be 1, 2 or 4.

(3) Give examples of three elliptic curves defined over Q where

the size of E(Q)[2] is 1, 2 and 4, respectively.

Exercise 2.12.7. Let Et :

y2+(1−t)xy−ty

=

x3−tx2

with t ∈ Q and

Δt =

t5(t2

− 11t − 1) = 0. As we saw in Example 2.5.4 (or Appendix

E), every curve Et has a subgroup isomorphic to Z/5Z. Use Sage to

find elliptic curves with torsion Z/5Z and rank 0, 1 and 2. Also, try

to find an elliptic curve Et with rank r, as high as possible. (Note:

the highest rank known — as of 6/1/2009 — for an elliptic curve with

Z/5Z torsion is 6, discovered by Dujella and Lecacheux in 2001; see

[Duj09].)

Exercise 2.12.8. Let p ≥ 2 be a prime and Ep :

y2

=

x3

+

p2.

Show

that there is no torsion point P ∈ Ep(Q) with y(P ) equal to

y = ±1,

±p2,

±3p,

±3p2,

or ± 3.

Prove that Q = (0,p) is a torsion point of exact order 3. Conclude

that {O,Q, 2Q} are the only torsion points on Ep(Q). (Note: for

p = 3, the point (−2,1) ∈ E3(Q). Show that it is not a torsion

point.)

Exercise 2.12.9. Prove Proposition 2.6.8, as follows:

(1) First show that if f(x) is a polynomial, f (x) its derivative,

and f(δ) = f (δ) = 0, then f(x) has a double root at δ.