Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(896\)\(\medspace = 2^{7} \cdot 7 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.1438646272.4 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.56.2t1.b.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.0.1568.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} + 4x^{6} - 8x^{5} + 14x^{4} - 8x^{3} + 4x^{2} - 8x - 2 \)
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The roots of $f$ are computed in $\Q_{ 191 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 15 + 36\cdot 191 + 75\cdot 191^{2} + 188\cdot 191^{3} + 2\cdot 191^{4} + 130\cdot 191^{5} +O(191^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 31 + 106\cdot 191 + 136\cdot 191^{2} + 165\cdot 191^{3} + 122\cdot 191^{4} + 97\cdot 191^{5} +O(191^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 33 + 146\cdot 191 + 90\cdot 191^{2} + 42\cdot 191^{3} + 73\cdot 191^{4} + 59\cdot 191^{5} +O(191^{6})\)
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| $r_{ 4 }$ | $=$ |
\( 52 + 187\cdot 191 + 59\cdot 191^{2} + 53\cdot 191^{3} + 81\cdot 191^{4} + 105\cdot 191^{5} +O(191^{6})\)
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| $r_{ 5 }$ | $=$ |
\( 69 + 6\cdot 191 + 81\cdot 191^{2} + 116\cdot 191^{3} + 52\cdot 191^{4} + 41\cdot 191^{5} +O(191^{6})\)
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| $r_{ 6 }$ | $=$ |
\( 95 + 52\cdot 191 + 110\cdot 191^{2} + 165\cdot 191^{3} + 174\cdot 191^{4} + 48\cdot 191^{5} +O(191^{6})\)
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| $r_{ 7 }$ | $=$ |
\( 120 + 45\cdot 191 + 167\cdot 191^{2} + 76\cdot 191^{3} + 178\cdot 191^{4} + 19\cdot 191^{5} +O(191^{6})\)
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| $r_{ 8 }$ | $=$ |
\( 162 + 183\cdot 191 + 42\cdot 191^{2} + 146\cdot 191^{3} + 77\cdot 191^{4} + 70\cdot 191^{5} +O(191^{6})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $-2$ |
| $4$ | $2$ | $(1,5)(2,3)(4,8)(6,7)$ | $0$ |
| $4$ | $2$ | $(1,6)(2,4)(7,8)$ | $0$ |
| $2$ | $4$ | $(1,4,2,6)(3,8,5,7)$ | $0$ |
| $2$ | $8$ | $(1,5,6,8,2,3,4,7)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,8,4,5,2,7,6,3)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.