| Name: | $C_2\times S_4$ |
| Order: | $48$ |
| Abelian: | no |
| Generators: | $\begin{bmatrix}\frac{i+1}{2}&\frac{i+1}{2}&0&0\\\frac{i-1}{2}&\frac{-i+1}{2}&0&0\\0&0&\frac{-i+1}{2}&\frac{-i+1}{2}\\0&0&\frac{-i-1}{2}&\frac{i+1}{2}\end{bmatrix}, \begin{bmatrix}0&1&0&0\\-1&0&0&0\\0&0&0&1\\0&0&-1&0\end{bmatrix}, \begin{bmatrix}\zeta_8&0&0&0\\0&\zeta_8^7&0&0\\0&0&\zeta_8^7&0\\0&0&0&\zeta_8\end{bmatrix}, \begin{bmatrix}0&0&0&1\\0&0&-1&0\\0&-1&0&0\\1&0&0&0\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$1$ |
$0$ |
$6$ |
$0$ |
$50$ |
$0$ |
$525$ |
$0$ |
$6426$ |
$0$ |
$86394$ |
| $a_2$ |
$1$ |
$1$ |
$3$ |
$7$ |
$26$ |
$96$ |
$432$ |
$2045$ |
$10432$ |
$55144$ |
$300548$ |
$1669515$ |
$9406817$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ |
$1$ |
$1$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ |
$3$ |
$3$ |
$6$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ |
$7$ |
$11$ |
$23$ |
$50$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ |
$26$ |
$45$ |
$99$ |
$225$ |
$525$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ |
$96$ |
$201$ |
$462$ |
$1090$ |
$2625$ |
$6426$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ |
$432$ |
$962$ |
$2291$ |
$5565$ |
$13727$ |
$34272$ |
$86394$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&1&0&0&0&0\\0&1&0&0&1&0&1&0&1&0\\0&0&2&0&0&0&0&2&0&3\\0&0&0&3&0&1&0&2&0&0\\0&1&0&0&3&0&2&0&4&0\\1&0&0&1&0&5&0&1&0&0\\0&1&0&0&2&0&3&0&4&0\\0&0&2&2&0&1&0&9&0&7\\0&1&0&0&4&0&4&0&9&0\\0&0&3&0&0&0&0&7&0&11\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&2&3&3&5&3&9&9&11\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-2$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ |
|---|
| $-$ | $1$ | $1/2$ | $1/48$ | $0$ | $1/8$ | $1/6$ | $3/16$ |
|---|
| $a_1=0$ | $11/16$ | $1/2$ | $1/48$ | $0$ | $1/8$ | $1/6$ | $3/16$ |
|---|
Additional information
The Sato-Tate group $J(O)$ has the largest component group (of order 48) among Sato-tate groups of abelian surfaces over number fields [10.1112/S0010437X12000279, arXiv:1110.6638].
