Basic invariants
| Dimension: | $4$ |
| Group: | $Q_8:C_2^2$ |
| Conductor: | \(3326976\)\(\medspace = 2^{10} \cdot 3^{2} \cdot 19^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.1916338176.1 |
| 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 16.0.478584585616890104119296.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 10x^{6} - 8x^{5} + 28x^{4} + 40x^{3} - 2x^{2} + 27 \)
|
The roots of $f$ are computed in $\Q_{ 73 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 9 + 10\cdot 73 + 2\cdot 73^{2} + 54\cdot 73^{3} + 19\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 10 + 13\cdot 73 + 40\cdot 73^{2} + 7\cdot 73^{3} + 51\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 14 + 34\cdot 73 + 64\cdot 73^{2} + 35\cdot 73^{3} + 12\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 17 + 36\cdot 73 + 6\cdot 73^{2} + 65\cdot 73^{3} + 34\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 25 + 37\cdot 73 + 67\cdot 73^{2} + 67\cdot 73^{3} + 36\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 33 + 65\cdot 73 + 72\cdot 73^{2} + 63\cdot 73^{3} + 5\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 43 + 58\cdot 73 + 35\cdot 73^{2} + 40\cdot 73^{3} + 32\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 68 + 36\cdot 73 + 2\cdot 73^{2} + 30\cdot 73^{3} + 25\cdot 73^{4} +O(73^{5})\)
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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 |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,3)(2,5)(4,6)(7,8)$ | $-4$ |
| $2$ | $2$ | $(1,6)(2,8)(3,4)(5,7)$ | $0$ |
| $2$ | $2$ | $(1,2)(3,5)(4,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,6)(3,7)(4,5)$ | $0$ |
| $2$ | $2$ | $(2,5)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,6)(2,7)(3,4)(5,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,5)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,6)(3,8)(4,5)$ | $0$ |
| $2$ | $2$ | $(1,2)(3,5)(4,8)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,4,3,6)(2,7,5,8)$ | $0$ |
| $2$ | $4$ | $(1,7,3,8)(2,4,5,6)$ | $0$ |
| $2$ | $4$ | $(1,5,3,2)(4,8,6,7)$ | $0$ |
| $2$ | $4$ | $(1,4,3,6)(2,8,5,7)$ | $0$ |
| $2$ | $4$ | $(1,8,3,7)(2,4,5,6)$ | $0$ |
| $2$ | $4$ | $(1,5,3,2)(4,7,6,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.