# gps_st downloaded from the LMFDB on 12 July 2026. # Search link: https://www.lmfdb.org/SatoTateGroup/?include_irrational=yes&component_group=[20,2] # Query "{'component_group': '20.2'}" returned 1 gps_st, sorted by weight. # Each entry in the following data list has the form: # [Label, Wt, Deg, $\mathrm{dim}_{\mathbb{R}}$, $\mathrm{G}^0$, Name, $\mathrm{G}/\mathrm{G}^0$, $\mathrm{Pr}[t\!=\!0]$, $\mathrm{E}[a_1^2]$, $\mathrm{E}[a_1^4]$, $\mathrm{E}[a_2]$] # For more details, see the definitions at the bottom of the file. "0.1.20" 0 1 0 "SO(1)" "\\mu(20)" 20 0 0 0 NULL # Label -- # In general Sato-Tate group labels have the form $w.d.A.c.ns$, where # * $w$ is the weight (nonnegative integer); # * $d$ is the degree (positive integer, even if $w$ is odd); # * $A$ is an uppercase letter that identifies the identity component among those of weight $w$ and degree $d$ (ordered by moment simplex); # * $c$ is the number of components (positive integer); # * $n$ is the second digit of the GAP id $[c,n]$ of the component group (positive integer); # * $s$ is a lowercase letter used to distinguish groups for which all the preceding invariants coincide (ordered by moment simplex). # When $w=0$ we the identity component is necessarily trivial and the $A$ is omitted. When $w=0$ and $d=1$ there is exactly one Sato-Tate group for each value of $c$ and we may omit $n$ and $s$. #Wt (weight) -- # The **weight** $w$ of the Sato-Tate group $G$ of a motive $X$ is determined by the cohomology group $H^w(X,\mathbb{Q}_\ell)$ used to define $G$. For a prime of norm $q$, the characteristic polynomial of Frobenius is a Weil $q^w$-polynomial. #Deg (degree) -- # 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. #$\mathrm{dim}_{\mathbb{R}}$ (real_dimension) -- # The **real dimension** of a Sato-Tate group is its dimension as a real Lie group. #$\mathrm{G}^0$ (identity_component) -- # The **identity component** of a Sato-Tate group $G$ is the connected component $G^0$ of the identity element. As a compact Lie group, the identity component $G^0$ is a normal subgroup of finite index. The quotient $G/G^0$ is the component group of $G$. #Name (pretty) -- # The names of the Sato-Tate groups of weight 1 and degree 4 are taken from \cite{arXiv:1110.6638,doi:10.1112/S0010437X12000279}. #$\mathrm{G}/\mathrm{G}^0$ (components) -- # The **component group** of a Sato-Tate group $G$ is the quotient $G/G^0$ of $G$ by its identity component $G^0$, which is a normal subgroup of finite index. # If $G$ is the Sato-Tate group of a motive defined over a number field $K$, then the component group $G/G^0$ is canonically isomorphic to the Galois group of a finite extension $L/K$. For the motive attached to an abelian variety $A$ of dimension at most 3, $L$ is the smallest field over which all endomorphisms of $A_{\overline{K}}$ (the base change of $A$ to an algebraic closure of $K$) are defined. # Component groups are named according to their isomorphism type. This does not determine them uniquely; one can obtain a more explicit description by examining the list of generators. Common notations used in component group names include: # * $C_n$, the cyclic group of order $n$; # * $D_n$, the dihedral group of order $2n$; # * $A_n$, the alternating group on $n$ letters; # * $S_n$, the symmetric group on $n$ letters. #$\mathrm{Pr}[t\!=\!0]$ (trace_zero_density) -- # The **trace zero density** of a Sato-Tate group is the probability that a randomly sampled element (chosen according to the Haar measure) has trace zero. # On each component of a Sato-Tate group, the trace is either constant and equal to zero, or varies continuously taking each particular value with probability zero; so this probability is simply the ratio of the number of components on which the trace is identically zero to the total number of components. #$\mathrm{E}[a_1^2]$ (second_trace_moment) -- # The **second trace moment** of a Sato-Tate group $G$ is the expected value of the square of the trace of a random element of $G$ (under its Haar measure). # When $G$ is the Sato-Tate group of an abelian variety, this is equal to the dimension of its endomorphism algebra \cite{arxiv:1910.00518,mr:4038255}. #$\mathrm{E}[a_1^4]$ (fourth_trace_moment) -- # The **fourth trace moment** of a Sato-Tate group $G$ is the expected value of the fourth power of the trace of a random element of $G$ (under the Haar measure). # By a result known as "Larsen's alternative" \cite{mr:2058618}, if the fourth trace moment is less than or equal to 5 then $G$ is irreducible, and for $G$ of odd weight, the fourth trace moment is 3 when $G=\mathrm{USp}(2g)$ and greater than 3 otherwise. #$\mathrm{E}[a_2]$ (first_a2_moment) -- # The **first $a_2$ moment** of a Sato-Tate group $G$ is the expected value of the quadratic coefficient of the characteristic polynomial of a random element of $G$ (under its Haar measure). # When $G$ is the Sato-Tate group of an abelian variety, the first $a_2$ moment is equal to the rank of its Néron–Severi group \cite{arxiv:1910.00518,mr:4038255}.