Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(548\)\(\medspace = 2^{2} \cdot 137 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.658266368.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.548.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.0.2192.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} + 12x^{6} - 18x^{5} + 17x^{4} - 6x^{3} + 2x^{2} + 12x + 4 \)
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The roots of $f$ are computed in $\Q_{ 173 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 89\cdot 173 + 101\cdot 173^{2} + 74\cdot 173^{3} + 111\cdot 173^{4} + 115\cdot 173^{5} +O(173^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 5 + 96\cdot 173 + 166\cdot 173^{2} + 35\cdot 173^{3} + 47\cdot 173^{4} + 42\cdot 173^{5} +O(173^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 76 + 84\cdot 173^{2} + 166\cdot 173^{3} + 20\cdot 173^{4} + 25\cdot 173^{5} +O(173^{6})\)
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| $r_{ 4 }$ | $=$ |
\( 78 + 7\cdot 173 + 149\cdot 173^{2} + 127\cdot 173^{3} + 129\cdot 173^{4} + 124\cdot 173^{5} +O(173^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 118 + 121\cdot 173 + 35\cdot 173^{2} + 61\cdot 173^{3} + 79\cdot 173^{4} + 153\cdot 173^{5} +O(173^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 128 + 158\cdot 173 + 67\cdot 173^{2} + 25\cdot 173^{3} + 173^{4} + 161\cdot 173^{5} +O(173^{6})\)
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| $r_{ 7 }$ | $=$ |
\( 139 + 90\cdot 173 + 27\cdot 173^{2} + 118\cdot 173^{3} + 103\cdot 173^{4} + 117\cdot 173^{5} +O(173^{6})\)
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| $r_{ 8 }$ | $=$ |
\( 149 + 127\cdot 173 + 59\cdot 173^{2} + 82\cdot 173^{3} + 25\cdot 173^{4} + 125\cdot 173^{5} +O(173^{6})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $-2$ |
| $4$ | $2$ | $(1,8)(2,6)(3,7)(4,5)$ | $0$ |
| $4$ | $2$ | $(1,4)(5,6)(7,8)$ | $0$ |
| $2$ | $4$ | $(1,3,4,2)(5,7,8,6)$ | $0$ |
| $2$ | $8$ | $(1,7,3,8,4,6,2,5)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,8,2,7,4,5,3,6)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.