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