2.5. The torsion subgroup 33
Curve Torsion Generators
y2
=
x3
− 2 trivial O
y2
=
x3
+ 8 Z/2Z (−2,0)
y2
=
x3
+ 4 Z/3Z (0,2)
y2
=
x3
+ 4x Z/4Z (2,4)
y2
− y =
x3
−
x2
Z/5Z (0,1)
y2
=
x3
+ 1 Z/6Z (2,3)
y2
=
x3
− 43x + 166 Z/7Z (3,8)
y2
+ 7xy =
x3
+ 16x Z/8Z (−2,10)
y2 + xy + y = x3 − x2 − 14x + 29 Z/9Z (3,1)
y2 + xy = x3 − 45x + 81 Z/10Z (0,9)
y2 + 43xy − 210y = x3 − 210x2 Z/12Z (0,210)
y2
=
x3
− 4x Z/2Z ⊕ Z/2Z
((2,0))
y2 = x3 + 2x2 − 3x Z/4Z ⊕ Z/2Z
((0,0))
(3,6)
(0,0)
y2 + 5xy − 6y = x3 − 3x2 Z/6Z ⊕ Z/2Z
(
(−3,18)
(2,−2)
)
y2 + 17xy − 120y = x3 − 60x2 Z/8Z ⊕ Z/2Z
(
(30,−90)
(−40,400)
)
Figure 4. Examples of each of the possible torsion subgroups
over Q.
Furthermore, it is known that, if G is any of the groups in Eq.
2.5, there are infinitely many elliptic curves whose torsion subgroup is
isomorphic to G. See, for example, [Kub76], Table 3, p. 217. For the
convenience of the reader, the table in Kubert’s article is reproduced
in Appendix E.
Example 2.5.4. Let Et :
y2
+ (1 − t)xy − ty =
x3
−
tx2
with t ∈ Q
and Δt =
t5(t2
− 11t − 1) = 0. Then, the torsion subgroup of Et(Q)
contains a subgroup isomorphic to Z/5Z, and (0,0) is a point of exact
order 5. Conversely, if E : y2 = x3 + Ax + B is an elliptic curve with
torsion subgroup equal to Z/5Z, then there is an invertible change of
variables that takes E to an equation of the form Et for some t ∈ Q.
See also Examples E.1.1 and E.1.2.
A useful and simple consequence of Mazur’s theorem is that if
the order of a rational point P ∈ E(Q) is larger than 12, then P must
be a point of infinite order and, therefore, E(Q) contains an infinite
