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.
