Basic invariants
| Dimension: | $4$ |
| Group: | $Q_8:C_2^2$ |
| Conductor: | \(1334025\)\(\medspace = 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 11^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.588305025.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $Q_8:C_2^2$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $C_2^4$ |
| Projective field: | Galois closure of 16.0.3167056956579818375390625.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} + 6x^{6} - x^{5} + 19x^{4} + 9x^{3} + 22x^{2} + 7x + 7 \)
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The roots of $f$ are computed in $\Q_{ 331 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 23 + 103\cdot 331 + 162\cdot 331^{2} + 278\cdot 331^{3} + 331^{4} +O(331^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 39 + 239\cdot 331 + 242\cdot 331^{2} + 236\cdot 331^{3} + 83\cdot 331^{4} +O(331^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 107 + 114\cdot 331 + 263\cdot 331^{2} + 136\cdot 331^{3} + 163\cdot 331^{4} +O(331^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 136 + 208\cdot 331 + 146\cdot 331^{2} + 189\cdot 331^{3} + 315\cdot 331^{4} +O(331^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 223 + 93\cdot 331 + 151\cdot 331^{2} + 329\cdot 331^{3} + 240\cdot 331^{4} +O(331^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 234 + 57\cdot 331 + 263\cdot 331^{2} + 268\cdot 331^{3} + 131\cdot 331^{4} +O(331^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 267 + 112\cdot 331 + 203\cdot 331^{2} + 271\cdot 331^{3} + 243\cdot 331^{4} +O(331^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 298 + 63\cdot 331 + 222\cdot 331^{2} + 274\cdot 331^{3} + 142\cdot 331^{4} +O(331^{5})\)
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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 value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,3)(2,6)(4,5)(7,8)$ | $-4$ |
| $2$ | $2$ | $(1,3)(2,6)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,8)(3,4)(6,7)$ | $0$ |
| $2$ | $2$ | $(2,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,6)(2,3)(4,8)(5,7)$ | $0$ |
| $2$ | $2$ | $(1,2)(3,6)(4,8)(5,7)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,5)(3,7)(4,6)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,5)(3,8)(4,6)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,7)(3,4)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,6,3,2)(4,8,5,7)$ | $0$ |
| $2$ | $4$ | $(1,2,3,6)(4,8,5,7)$ | $0$ |
| $2$ | $4$ | $(1,8,3,7)(2,4,6,5)$ | $0$ |
| $2$ | $4$ | $(1,7,3,8)(2,4,6,5)$ | $0$ |
| $2$ | $4$ | $(1,5,3,4)(2,7,6,8)$ | $0$ |
| $2$ | $4$ | $(1,5,3,4)(2,8,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.