Basic invariants
| Dimension: | $4$ |
| Group: | $Q_8:C_2^2$ |
| Conductor: | \(2073600\)\(\medspace = 2^{10} \cdot 3^{4} \cdot 5^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.466560000.1 |
| 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 16.0.11007531417600000000.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 6x^{6} + 27x^{4} - 54x^{2} + 36 \)
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The roots of $f$ are computed in $\Q_{ 241 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 11 + 133\cdot 241 + 12\cdot 241^{2} + 67\cdot 241^{3} + 81\cdot 241^{4} + 134\cdot 241^{5} +O(241^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 28 + 99\cdot 241 + 117\cdot 241^{2} + 32\cdot 241^{3} + 80\cdot 241^{4} + 168\cdot 241^{5} +O(241^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 72 + 13\cdot 241 + 192\cdot 241^{2} + 186\cdot 241^{3} + 81\cdot 241^{4} + 29\cdot 241^{5} +O(241^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 119 + 21\cdot 241 + 150\cdot 241^{2} + 129\cdot 241^{3} + 128\cdot 241^{4} + 173\cdot 241^{5} +O(241^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 122 + 219\cdot 241 + 90\cdot 241^{2} + 111\cdot 241^{3} + 112\cdot 241^{4} + 67\cdot 241^{5} +O(241^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 169 + 227\cdot 241 + 48\cdot 241^{2} + 54\cdot 241^{3} + 159\cdot 241^{4} + 211\cdot 241^{5} +O(241^{6})\)
|
| $r_{ 7 }$ | $=$ |
\( 213 + 141\cdot 241 + 123\cdot 241^{2} + 208\cdot 241^{3} + 160\cdot 241^{4} + 72\cdot 241^{5} +O(241^{6})\)
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| $r_{ 8 }$ | $=$ |
\( 230 + 107\cdot 241 + 228\cdot 241^{2} + 173\cdot 241^{3} + 159\cdot 241^{4} + 106\cdot 241^{5} +O(241^{6})\)
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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 |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-4$ |
| $2$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,6)(2,5)(3,8)(4,7)$ | $0$ |
| $2$ | $2$ | $(1,8)(4,5)$ | $0$ |
| $2$ | $2$ | $(1,8)(3,6)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ |
| $2$ | $2$ | $(1,6)(2,4)(3,8)(5,7)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,3)(4,8)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $0$ |
| $2$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $0$ |
| $2$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $0$ |
| $2$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ |
| $2$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $0$ |
| $2$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.