The **degree** $d$ of a Sato-Tate group is the degree of the characteristic polynomials of its elements, equivalently, the dimension of the $d\times d$ matrices it contains.

For an abelian variety $A$ over a number field, the degree $d$ of its Sato-Tate group is twice its dimension $g$ as an abelian variety (if $A=\mathrm{Jac}(C)$ is the Jacobian of a curve $C$, then $g$ is also the genus of $C$). The degree $d=2g$ is then also the degree of the characteristic polynomials of the Frobenius endomorphism of the reductions of $A$ modulo good primes.

For Artin motives, the degree of the Sato-Tate group is the same as the degree of the Artin representation (as a permutation group).

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by Kiran S. Kedlaya on 2018-06-20 04:06:54

**Referred to by:**

- rcs.cande.st_group
- rcs.rigor.st_group
- st_group.definition
- st_group.hodge_circle
- st_group.invariants
- st_group.label
- st_group.moment_simplex
- st_group.rational
- st_group.summary
- lmfdb/sato_tate_groups/main.py (line 824)
- lmfdb/sato_tate_groups/main.py (line 968)
- lmfdb/sato_tate_groups/templates/st_browse.html (line 15)
- lmfdb/sato_tate_groups/templates/st_display.html (line 7)
- lmfdb/sato_tate_groups/templates/st_results.html (line 13)

**History:**(expand/hide all)

- 2021-01-01 15:19:50 by Andrew Sutherland
- 2021-01-01 15:19:25 by Andrew Sutherland
- 2018-06-20 04:06:54 by Kiran S. Kedlaya (Reviewed)

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