Some Data on Torsion Subgroups of Elliptic Curves

Javier Fernandez

Department of Mathematics
University of Utah
Salt Lake City
UT 84112

jfernand@math.utah.edu

A well known theorem of B. Mazur [1] states that the only possible groups that appear as torsion subgroups of elliptic curves over $\ensuremath{\mathbb{Q}}$ are of the form

$$\ensuremath{\mathbb{Z}}_d d=1,2,3,4,5,6,7,8,9,10,12$$

where $\ensuremath{\mathbb{Z}}_d = \ensuremath{\mathbb{Z}}/(d\ensuremath{\mathbb{Z}})$ or

$$\ensuremath{\mathbb{Z}}_2 \times \ensuremath{\mathbb{Z}}_2 d=2,4,6,8.$$

Table 1 shows, for each possible group, an elliptic curve that realizes the group, as well as a set of generators. All curves come from [2], Exercise 2.12.

Table 2 shows the frequency of each group over different regions of the space of elliptic curves. Two realizations of the elliptic curves are used. The usual Weierstrass form

$$y^2=x^3+a x^2 +b x+c,$$ (1)

and the reduced minimal model form:

$$y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6 a_1, a_3\in\{0,1\},\, a_2\in \{-1,0,1\}.$$ (2)

^{1 2}

Table 2: Torsion group frequencies

group	frequency
	W I3	W II4	M I5	M II6
$\ensuremath{\mathbb{Z}}_1$	$0.9817$	$0.9907113$	$0.9968541$	$NC$
$\ensuremath{\mathbb{Z}}_2$	$0.0177$	$0.0090987$	$0.0030424$	$NC$
$\ensuremath{\mathbb{Z}}_3$	$0.00018$	$6.177\, 10^{-5}$	$5.819\, 10^{-5}$	$NC$
$\ensuremath{\mathbb{Z}}_4$	$8.8\, 10^{-5}$	$2.551\, 10^{-5}$	$1.248\, 10^{-5}$	$NC$
$\ensuremath{\mathbb{Z}}_5$	$3.8\, 10^{-6}$	$8.594\, 10^{-7}$	$1.378\, 10^{-6}$	$NC$
$\ensuremath{\mathbb{Z}}_6$	$5.0\, 10^{-7}$	$5.625\, 10^{-6}$	$2.730\, 10^{-6}$	$NC$
$\ensuremath{\mathbb{Z}}_7$	$5.0\, 10^{-7}$	$1.406\, 10^{-7}$	$1.560\, 10^{-7}$	$NC$
$\ensuremath{\mathbb{Z}}_8$	$6.2\, 10^{-7}$	$2.188\, 10^{-7}$	$1.820\, 10^{-7}$	$NC$
$\ensuremath{\mathbb{Z}}_9$	0	0	$5.201\, 10^{-8}$	$2.08\, 10^{-9}$
$\ensuremath{\mathbb{Z}}_{10}$	0	0	$7.801\, 10^{-8}$	$5.21\, 10^{-9}$
$\ensuremath{\mathbb{Z}}_{12}$	0	0	$2.600\, 10^{-8}$	$1.04\, 10^{-9}$
$\ensuremath{\mathbb{Z}}_2\times \ensuremath{\mathbb{Z}}_2$	$0.00029$	$9.363\, 10^{-5}$	$2.741\, 10^{-5}$	$NC$
$\ensuremath{\mathbb{Z}}_4\times \ensuremath{\mathbb{Z}}_2$	$8.5\, 10^{-6}$	$2.078\, 10^{-6}$	$7.021\, 10^{-7}$	$NC$
$\ensuremath{\mathbb{Z}}_6\times \ensuremath{\mathbb{Z}}_2$	0	$1.094\, 10^{-7}$	$5.201\, 10^{-8}$	$4.17\, 10^{-9}$
$\ensuremath{\mathbb{Z}}_8\times \ensuremath{\mathbb{Z}}_2$	0	0	0	$1.04\, 10^{-9}$