⌂
→
Sato-Tate groups
→
Search results
Citation
·
Feedback
·
Hide Menu
Sato-Tate group search results
Introduction
Overview
Random
Universe
Knowledge
L-functions
Rational
All
Modular forms
Classical
Maass
Hilbert
Bianchi
Varieties
Elliptic curves over
Q
\Q
Q
Elliptic curves over
Q
(
α
)
\Q(\alpha)
Q
(
α
)
Genus 2 curves over
Q
\Q
Q
Higher genus families
Abelian varieties over
F
q
\F_{q}
F
q
Fields
Number fields
p
p
p
-adic fields
Representations
Dirichlet characters
Artin representations
Groups
Galois groups
Sato-Tate groups
Database
↑
Learn more
Source and acknowledgments
Completeness of the data
Reliability of the data
Sato-Tate group labels
Refine search
Weight
Degree
Include irrational
S
T
0
\mathrm{ST}^0
S
T
0
Maximal
no
yes
SO(1)
SO(2)
SO(3)
SO(4)
SO(5)
SO(6)
U(1)
SU(2)
U(1)_2
SU(2)_2
U(1)xU(1)
U(1)xSU(2)
SU(2)xSU(2)
USp(4)
U(1)_3
SU(2)_3
U(1)xU(1)_2
SU(2)xU(1)_2
U(1)xSU(2)_2
SU(2)xSU(2)_2
U(1)^3
U(1)^2xSU(2)
U(1)xSU(2)^2
SU(2)^3
U(1)xUSp(4)
SU(2)xUSp(4)
U(3)
USp(6)
yes
no
Components
Component group
P
r
[
a
1
=
0
]
\mathrm{Pr}[a_1=0]
P
r
[
a
1
=
0
]
Second trace moment
Fourth trace moment
First
a
2
a_2
a
2
moment
Sort order
Select
Search again
Random group
▲ weight
degree
real dimension
component group
trace zero density
character diagonal
columns to display
✓ label
✓ weight
✓ degree
✓ real dimension
✓ identity component
✓ name
✓ components
✓ Pr[t=0]
✓ E[a_1^2]
✓ E[a_1^4]
E[a_1^6]
E[a_1^8]
E[a_1^{10}]
E[a_1^{12}]
✓ E[a_2]
E[a_2^2]
E[a_2^3]
E[a_2^4]
E[a_2^5]
E[a_2^6]
E[a_3^2]
E[a_3^4]
E[a_3^6]
maximal
rational
diagonal
show all
Results (unique match)
Download
displayed columns
for
results
to
Text
Pari/GP
SageMath
Magma
Oscar
CSV
Label
Wt
Deg
d
i
m
R
\mathrm{dim}_{\mathbb{R}}
d
i
m
R
G
0
\mathrm{G}^0
G
0
Name
G
/
G
0
\mathrm{G}/\mathrm{G}^0
G
/
G
0
P
r
[
t
=
0
]
\mathrm{Pr}[t\!=\!0]
P
r
[
t
=
0
]
E
[
a
1
2
]
\mathrm{E}[a_1^2]
E
[
a
1
2
]
E
[
a
1
4
]
\mathrm{E}[a_1^4]
E
[
a
1
4
]
E
[
a
1
6
]
\mathrm{E}[a_1^6]
E
[
a
1
6
]
E
[
a
1
8
]
\mathrm{E}[a_1^8]
E
[
a
1
8
]
E
[
a
1
10
]
\mathrm{E}[a_1^{10}]
E
[
a
1
1
0
]
E
[
a
1
12
]
\mathrm{E}[a_1^{12}]
E
[
a
1
1
2
]
E
[
a
2
]
\mathrm{E}[a_2]
E
[
a
2
]
E
[
a
2
2
]
\mathrm{E}[a_2^2]
E
[
a
2
2
]
E
[
a
2
3
]
\mathrm{E}[a_2^3]
E
[
a
2
3
]
E
[
a
2
4
]
\mathrm{E}[a_2^4]
E
[
a
2
4
]
E
[
a
2
5
]
\mathrm{E}[a_2^5]
E
[
a
2
5
]
E
[
a
2
6
]
\mathrm{E}[a_2^6]
E
[
a
2
6
]
E
[
a
3
2
]
\mathrm{E}[a_3^2]
E
[
a
3
2
]
E
[
a
3
4
]
\mathrm{E}[a_3^4]
E
[
a
3
4
]
E
[
a
3
6
]
\mathrm{E}[a_3^6]
E
[
a
3
6
]
Maximal
Rational
Diagonal
1.6.N.432.523a
1
1
1
6
6
6
1
1
1
U
(
1
)
3
\mathrm{U}(1)_{3}
U
(
1
)
3
J
(
D
(
6
,
6
)
)
J(D(6,6))
J
(
D
(
6
,
6
)
)
432
432
4
3
2
145
/
216
145/216
1
4
5
/
2
1
6
1
1
1
9
9
9
160
160
1
6
0
3745
3745
3
7
4
5
97776
97776
9
7
7
7
6
97776
97776
9
7
7
7
6
1
1
1
4
4
4
21
21
2
1
170
170
1
7
0
1731
1731
1
7
3
1
19643
19643
1
9
6
4
3
4
4
4
420
420
4
2
0
92540
92540
9
2
5
4
0
✓
✓
[
1
,
1
,
3
,
3
,
5
,
13
,
17
,
25
,
31
,
22
,
10
,
64
,
82
,
160
,
301
,
154
,
158
,
352
,
287
,
102
]
[1, 1, 3, 3, 5, 13, 17, 25, 31, 22, 10, 64, 82, 160, 301, 154, 158, 352, 287, 102]
[
1
,
1
,
3
,
3
,
5
,
1
3
,
1
7
,
2
5
,
3
1
,
2
2
,
1
0
,
6
4
,
8
2
,
1
6
0
,
3
0
1
,
1
5
4
,
1
5
8
,
3
5
2
,
2
8
7
,
1
0
2
]
Download
displayed columns
for
results
to
Text
Pari/GP
SageMath
Magma
Oscar
CSV