Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(495\)\(\medspace = 3^{2} \cdot 5 \cdot 11 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.1334161125.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.55.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.2.2475.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} + 6x^{6} + x^{5} + 4x^{4} + 3x^{3} + 6x^{2} + 3x + 3 \)
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The roots of $f$ are computed in $\Q_{ 179 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 9\cdot 179 + 153\cdot 179^{2} + 106\cdot 179^{3} + 52\cdot 179^{4} + 127\cdot 179^{5} +O(179^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 23 + 85\cdot 179 + 70\cdot 179^{2} + 33\cdot 179^{3} + 107\cdot 179^{4} + 129\cdot 179^{5} +O(179^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 31 + 139\cdot 179 + 53\cdot 179^{2} + 19\cdot 179^{3} + 171\cdot 179^{4} + 140\cdot 179^{5} +O(179^{6})\)
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| $r_{ 4 }$ | $=$ |
\( 39 + 72\cdot 179 + 102\cdot 179^{2} + 104\cdot 179^{3} + 101\cdot 179^{4} + 83\cdot 179^{5} +O(179^{6})\)
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| $r_{ 5 }$ | $=$ |
\( 87 + 61\cdot 179 + 131\cdot 179^{2} + 21\cdot 179^{3} + 157\cdot 179^{4} + 3\cdot 179^{5} +O(179^{6})\)
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| $r_{ 6 }$ | $=$ |
\( 108 + 137\cdot 179 + 48\cdot 179^{2} + 127\cdot 179^{3} + 32\cdot 179^{4} + 6\cdot 179^{5} +O(179^{6})\)
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| $r_{ 7 }$ | $=$ |
\( 113 + 101\cdot 179 + 28\cdot 179^{2} + 38\cdot 179^{3} + 14\cdot 179^{4} + 128\cdot 179^{5} +O(179^{6})\)
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| $r_{ 8 }$ | $=$ |
\( 136 + 109\cdot 179 + 127\cdot 179^{2} + 85\cdot 179^{3} + 79\cdot 179^{4} + 96\cdot 179^{5} +O(179^{6})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)(7,8)$ | $-2$ |
| $4$ | $2$ | $(2,5)(3,8)(4,7)$ | $0$ |
| $4$ | $2$ | $(1,4)(2,8)(3,6)(5,7)$ | $0$ |
| $2$ | $4$ | $(1,5,6,2)(3,7,4,8)$ | $0$ |
| $2$ | $8$ | $(1,4,5,8,6,3,2,7)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,8,2,4,6,7,5,3)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.