Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(164\)\(\medspace = 2^{2} \cdot 41 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.180848704.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.164.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.0.656.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} + x^{6} + 4x^{5} - 4x^{4} + 4x^{3} + x^{2} - 3x + 1 \)
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The roots of $f$ are computed in $\Q_{ 173 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 6 + 110\cdot 173 + 82\cdot 173^{2} + 73\cdot 173^{3} + 62\cdot 173^{4} +O(173^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 9 + 114\cdot 173 + 100\cdot 173^{2} + 123\cdot 173^{3} + 153\cdot 173^{4} +O(173^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 12 + 171\cdot 173 + 73\cdot 173^{2} + 145\cdot 173^{3} + 93\cdot 173^{4} +O(173^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 25 + 61\cdot 173 + 17\cdot 173^{2} + 63\cdot 173^{3} + 3\cdot 173^{4} +O(173^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 29 + 16\cdot 173 + 28\cdot 173^{2} + 122\cdot 173^{3} + 51\cdot 173^{4} +O(173^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 77 + 43\cdot 173 + 74\cdot 173^{2} + 119\cdot 173^{3} + 103\cdot 173^{4} +O(173^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 90 + 29\cdot 173 + 5\cdot 173^{2} + 128\cdot 173^{3} + 131\cdot 173^{4} +O(173^{5})\)
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| $r_{ 8 }$ | $=$ |
\( 101 + 146\cdot 173 + 136\cdot 173^{2} + 89\cdot 173^{3} + 91\cdot 173^{4} +O(173^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,6)(3,8)(4,7)$ | $-2$ |
| $4$ | $2$ | $(1,7)(2,3)(4,5)(6,8)$ | $0$ |
| $4$ | $2$ | $(2,7)(3,8)(4,6)$ | $0$ |
| $2$ | $4$ | $(1,8,5,3)(2,4,6,7)$ | $0$ |
| $2$ | $8$ | $(1,2,8,4,5,6,3,7)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,4,3,2,5,7,8,6)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.