The database currently contains 55 Sato-Tate groups. The statistics below omit the infinite family $\mu(n)$ with trivial identity component since they are generated dynamically in search results.

## Distribution of component groups and identity components

identity component
$\operatorname{SU}(2)$ $\operatorname{SU}(2)_2$ $\operatorname{SU}(2)\times\operatorname{SU}(2)$ $\operatorname{U}(1)$ $\operatorname{U}(1)_2$ $\operatorname{U}(1)\times\operatorname{SU}(2)$ $\operatorname{U}(1)\times\operatorname{U}(1)$ $\operatorname{USp}(4)$ Total
component group
$C_1$ 1 1 1 1 1 1 1 1 8
100.00% 10.00% 50.00% 50.00% 3.12% 50.00% 20.00% 100.00% 14.55%
$C_2$ 2 1 1 3 1 2 10
20.00% 50.00% 50.00% 9.38% 50.00% 40.00% 18.18%
$C_3$ 1 1 2
10.00% 3.12% 3.64%
$C_4$ 1 2 1 4
10.00% 6.25% 20.00% 7.27%
$C_2^2$ 1 3 1 5
10.00% 9.38% 20.00% 9.09%
$S_3$ 1 2 3
10.00% 6.25% 5.45%
$C_6$ 1 3 4
10.00% 9.38% 7.27%
$C_2\times C_4$ 1 1
3.12% 1.82%
$D_4$ 1 3 4
10.00% 9.38% 7.27%
$C_2^3$ 1 1
3.12% 1.82%
$A_4$ 1 1
3.12% 1.82%
$D_6$ 1 4 5
10.00% 12.50% 9.09%
$C_2\times C_6$ 1 1
3.12% 1.82%
$C_2\times D_4$ 1 1
3.12% 1.82%
$S_4$ 2 2
6.25% 3.64%
$C_2\times A_4$ 1 1
3.12% 1.82%
$C_2^2\times S_3$ 1 1
3.12% 1.82%
$C_2\times S_4$ 1 1
3.12% 1.82%
Total 1 10 2 2 32 2 5 1 55
1.82% 18.18% 3.64% 3.64% 58.18% 3.64% 9.09% 1.82%

## Distribution of maximal subgroups per identity component

 identity component count proportion $\operatorname{SU}(2)$ $\operatorname{SU}(2)_2$ $\operatorname{SU}(2)\times\operatorname{SU}(2)$ $\operatorname{U}(1)$ $\operatorname{U}(1)_2$ $\operatorname{U}(1)\times\operatorname{SU}(2)$ $\operatorname{U}(1)\times\operatorname{U}(1)$ $\operatorname{USp}(4)$ 1 2 1 1 2 1 3 1 8.33% 16.67% 8.33% 8.33% 16.67% 8.33% 25.00% 8.33%

## Distribution of trace zero densities and identity components

identity component
$\operatorname{SU}(2)$ $\operatorname{SU}(2)_2$ $\operatorname{SU}(2)\times\operatorname{SU}(2)$ $\operatorname{U}(1)$ $\operatorname{U}(1)_2$ $\operatorname{U}(1)\times\operatorname{SU}(2)$ $\operatorname{U}(1)\times\operatorname{U}(1)$ $\operatorname{USp}(4)$ Total
trace zero density
0 1 2 1 1 2 2 2 1 12
100.00% 20.00% 50.00% 50.00% 6.25% 100.00% 40.00% 100.00% 21.82%
1/6 1 1 2
10.00% 3.12% 3.64%
1/4 1 2 1 4
10.00% 6.25% 20.00% 7.27%
3/8 1 1
3.12% 1.82%
1/2 3 1 1 7 1 13
30.00% 50.00% 50.00% 21.88% 20.00% 23.64%
7/12 1 3 4
10.00% 9.38% 7.27%
5/8 1 5 6
10.00% 15.62% 10.91%
11/16 1 1
3.12% 1.82%
3/4 1 6 1 8
10.00% 18.75% 20.00% 14.55%
19/24 1 1
3.12% 1.82%
13/16 1 1
3.12% 1.82%
7/8 2 2
6.25% 3.64%
Total 1 10 2 2 32 2 5 1 55
1.82% 18.18% 3.64% 3.64% 58.18% 3.64% 9.09% 1.82%

## Distribution of second trace moments and identity components

identity component
$\operatorname{SU}(2)$ $\operatorname{SU}(2)_2$ $\operatorname{SU}(2)\times\operatorname{SU}(2)$ $\operatorname{U}(1)$ $\operatorname{U}(1)_2$ $\operatorname{U}(1)\times\operatorname{SU}(2)$ $\operatorname{U}(1)\times\operatorname{U}(1)$ $\operatorname{USp}(4)$ Total
second trace moment
1 1 4 1 1 9 1 1 18
100.00% 40.00% 50.00% 50.00% 28.12% 20.00% 100.00% 32.73%
2 5 1 1 16 1 2 26
50.00% 50.00% 50.00% 50.00% 50.00% 40.00% 47.27%
3 1 1 2
50.00% 20.00% 3.64%
4 1 6 1 8
10.00% 18.75% 20.00% 14.55%
8 1 1
3.12% 1.82%
Total 1 10 2 2 32 2 5 1 55
1.82% 18.18% 3.64% 3.64% 58.18% 3.64% 9.09% 1.82%

## Distribution of fourth trace moments and identity components

identity component
$\operatorname{SU}(2)$ $\operatorname{SU}(2)_2$ $\operatorname{SU}(2)\times\operatorname{SU}(2)$ $\operatorname{U}(1)$ $\operatorname{U}(1)_2$ $\operatorname{U}(1)\times\operatorname{SU}(2)$ $\operatorname{U}(1)\times\operatorname{U}(1)$ $\operatorname{USp}(4)$ Total
fourth trace moment
2 1 1
100.00% 1.82%
3 1 1 2
50.00% 100.00% 3.64%
5 1 1
50.00% 1.82%
6 3 1 3 7
30.00% 50.00% 9.38% 12.73%
8 1 1
10.00% 1.82%
9 4 1 5
12.50% 20.00% 9.09%
10 1 1
50.00% 1.82%
11 1 1
50.00% 1.82%
12 3 4 1 8
30.00% 12.50% 20.00% 14.55%
16 2 2
20.00% 3.64%
18 10 1 11
31.25% 20.00% 20.00%
20 1 1
50.00% 1.82%
21 1 1
20.00% 1.82%
24 4 4
12.50% 7.27%
32 1 1
10.00% 1.82%
36 3 1 4
9.38% 20.00% 7.27%
48 3 3
9.38% 5.45%
96 1 1
3.12% 1.82%
Total 1 10 2 2 32 2 5 1 55
1.82% 18.18% 3.64% 3.64% 58.18% 3.64% 9.09% 1.82%

## Distribution of first $a_2$ moment and identity components

identity component
$\operatorname{SU}(2)_2$ $\operatorname{SU}(2)\times\operatorname{SU}(2)$ $\operatorname{U}(1)_2$ $\operatorname{U}(1)\times\operatorname{SU}(2)$ $\operatorname{U}(1)\times\operatorname{U}(1)$ $\operatorname{USp}(4)$ Total
first a2 moment
1 8 1 22 1 1 33
80.00% 50.00% 68.75% 20.00% 100.00% 63.46%
2 1 1 8 2 4 16
10.00% 50.00% 25.00% 100.00% 80.00% 30.77%
3 1 1 2
10.00% 3.12% 3.85%
4 1 1
3.12% 1.92%
Total 10 2 32 2 5 1 52
19.23% 3.85% 61.54% 3.85% 9.62% 1.92%