Basic invariants
| Dimension: | $4$ |
| Group: | $Q_8:C_2^2$ |
| Conductor: | \(921600\)\(\medspace = 2^{12} \cdot 3^{2} \cdot 5^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 8.0.368640000.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $Q_8:C_2^2$ |
| Parity: | even |
| Projective image: | $C_2^4$ |
| Projective field: | Galois closure of 16.0.11007531417600000000.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 409 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 30 + 353\cdot 409 + 252\cdot 409^{2} + 199\cdot 409^{3} + 383\cdot 409^{4} +O(409^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 90 + 14\cdot 409 + 238\cdot 409^{2} + 285\cdot 409^{3} + 333\cdot 409^{4} +O(409^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 133 + 26\cdot 409 + 206\cdot 409^{2} + 273\cdot 409^{3} + 364\cdot 409^{4} +O(409^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 203 + 172\cdot 409 + 330\cdot 409^{2} + 219\cdot 409^{3} + 186\cdot 409^{4} +O(409^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 206 + 236\cdot 409 + 78\cdot 409^{2} + 189\cdot 409^{3} + 222\cdot 409^{4} +O(409^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 276 + 382\cdot 409 + 202\cdot 409^{2} + 135\cdot 409^{3} + 44\cdot 409^{4} +O(409^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 319 + 394\cdot 409 + 170\cdot 409^{2} + 123\cdot 409^{3} + 75\cdot 409^{4} +O(409^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 379 + 55\cdot 409 + 156\cdot 409^{2} + 209\cdot 409^{3} + 25\cdot 409^{4} +O(409^{5})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 8 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 8 }$ | Character values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-4$ |
| $2$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ |
| $2$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,8)(4,5)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,3)(4,8)(6,7)$ | $0$ |
| $2$ | $2$ | $(1,8)(3,6)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ |
| $2$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ |
| $2$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ |
| $2$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $0$ |
| $2$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $0$ |
| $2$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $0$ |