Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(579\)\(\medspace = 3 \cdot 193 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.582313617.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.579.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.0.1737.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} + 6x^{6} - 15x^{5} + 29x^{4} - 30x^{3} + 24x^{2} - 24x + 16 \)
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The roots of $f$ are computed in $\Q_{ 151 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 4 + 138\cdot 151 + 113\cdot 151^{2} + 53\cdot 151^{3} + 32\cdot 151^{4} +O(151^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 27 + 69\cdot 151 + 73\cdot 151^{2} + 134\cdot 151^{3} + 121\cdot 151^{4} +O(151^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 56 + 97\cdot 151 + 136\cdot 151^{2} + 93\cdot 151^{3} + 121\cdot 151^{4} +O(151^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 71 + 2\cdot 151 + 68\cdot 151^{2} + 78\cdot 151^{3} + 132\cdot 151^{4} +O(151^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 76 + 20\cdot 151 + 90\cdot 151^{2} + 60\cdot 151^{3} + 70\cdot 151^{4} +O(151^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 113 + 105\cdot 151 + 4\cdot 151^{2} + 87\cdot 151^{3} + 108\cdot 151^{4} +O(151^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 117 + 81\cdot 151 + 68\cdot 151^{2} + 80\cdot 151^{3} + 141\cdot 151^{4} +O(151^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 143 + 88\cdot 151 + 48\cdot 151^{2} + 15\cdot 151^{3} + 26\cdot 151^{4} +O(151^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,3)(4,7)(6,8)$ | $-2$ |
| $4$ | $2$ | $(1,8)(4,7)(5,6)$ | $0$ |
| $4$ | $2$ | $(1,4)(2,8)(3,6)(5,7)$ | $0$ |
| $2$ | $4$ | $(1,8,5,6)(2,4,3,7)$ | $0$ |
| $2$ | $8$ | $(1,2,8,4,5,3,6,7)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,4,6,2,5,7,8,3)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.