Basic invariants
| Dimension: | $4$ |
| Group: | $Q_8:C_2^2$ |
| Conductor: | \(2433600\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 13^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.4.10281960000.1 |
| 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 16.0.35074927889488281600000000.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{6} - 8x^{5} - 17x^{4} + 4x^{3} + 49x^{2} + 4x - 16 \)
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The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 9 + 65\cdot 79 + 73\cdot 79^{2} + 53\cdot 79^{3} + 22\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 16 + 67\cdot 79 + 4\cdot 79^{2} + 7\cdot 79^{3} + 6\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 24 + 11\cdot 79 + 65\cdot 79^{2} + 41\cdot 79^{3} + 19\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 25 + 43\cdot 79 + 62\cdot 79^{2} + 79^{3} + 49\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 41 + 28\cdot 79 + 3\cdot 79^{2} + 74\cdot 79^{3} + 49\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 56 + 32\cdot 79 + 61\cdot 79^{2} + 44\cdot 79^{3} + 75\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 68 + 16\cdot 79 + 39\cdot 79^{2} + 57\cdot 79^{3} + 10\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 77 + 50\cdot 79 + 5\cdot 79^{2} + 35\cdot 79^{3} + 3\cdot 79^{4} +O(79^{5})\)
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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 |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,4)(2,8)(3,5)(6,7)$ | $-4$ |
| $2$ | $2$ | $(1,6)(2,3)(4,7)(5,8)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,4)(3,6)(5,7)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,8)$ | $0$ |
| $2$ | $2$ | $(1,2)(3,6)(4,8)(5,7)$ | $0$ |
| $2$ | $2$ | $(1,4)(3,5)$ | $0$ |
| $2$ | $2$ | $(2,8)(3,5)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,3)(4,6)(5,8)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,6)(3,4)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,6)(4,5)(7,8)$ | $0$ |
| $2$ | $4$ | $(1,6,4,7)(2,5,8,3)$ | $0$ |
| $2$ | $4$ | $(1,3,4,5)(2,6,8,7)$ | $0$ |
| $2$ | $4$ | $(1,2,4,8)(3,7,5,6)$ | $0$ |
| $2$ | $4$ | $(1,8,4,2)(3,7,5,6)$ | $0$ |
| $2$ | $4$ | $(1,5,4,3)(2,6,8,7)$ | $0$ |
| $2$ | $4$ | $(1,7,4,6)(2,5,8,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.