Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(2004\)\(\medspace = 2^{2} \cdot 3 \cdot 167 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.96577152768.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.2004.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.2.24048.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} + x^{6} + 8x^{5} - 12x^{4} - 8x^{3} - 44x^{2} + 16x - 32 \)
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The roots of $f$ are computed in $\Q_{ 251 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 4 + 158\cdot 251 + 112\cdot 251^{2} + 89\cdot 251^{3} + 69\cdot 251^{4} + 132\cdot 251^{5} +O(251^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 41 + 95\cdot 251 + 145\cdot 251^{2} + 248\cdot 251^{3} + 74\cdot 251^{4} + 228\cdot 251^{5} +O(251^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 50 + 220\cdot 251 + 195\cdot 251^{2} + 190\cdot 251^{3} + 198\cdot 251^{4} + 162\cdot 251^{5} +O(251^{6})\)
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| $r_{ 4 }$ | $=$ |
\( 116 + 166\cdot 251 + 44\cdot 251^{2} + 81\cdot 251^{3} + 113\cdot 251^{4} + 212\cdot 251^{5} +O(251^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 168 + 209\cdot 251 + 241\cdot 251^{2} + 107\cdot 251^{3} + 144\cdot 251^{4} + 240\cdot 251^{5} +O(251^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 193 + 158\cdot 251 + 97\cdot 251^{2} + 190\cdot 251^{3} + 13\cdot 251^{4} + 122\cdot 251^{5} +O(251^{6})\)
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| $r_{ 7 }$ | $=$ |
\( 213 + 251 + 137\cdot 251^{2} + 112\cdot 251^{3} + 61\cdot 251^{4} + 115\cdot 251^{5} +O(251^{6})\)
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| $r_{ 8 }$ | $=$ |
\( 223 + 244\cdot 251 + 28\cdot 251^{2} + 234\cdot 251^{3} + 76\cdot 251^{4} + 41\cdot 251^{5} +O(251^{6})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)(7,8)$ | $-2$ |
| $4$ | $2$ | $(1,8)(2,3)(4,7)(5,6)$ | $0$ |
| $4$ | $2$ | $(1,5)(2,4)(7,8)$ | $0$ |
| $2$ | $4$ | $(1,2,4,5)(3,8,6,7)$ | $0$ |
| $2$ | $8$ | $(1,7,2,3,4,8,5,6)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,3,5,7,4,6,2,8)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.