Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(95\)\(\medspace = 5 \cdot 19 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.16290125.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.95.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.2.475.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} - x^{6} + 4x^{5} + x^{4} - 4x^{3} - x^{2} + x + 1 \)
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The roots of $f$ are computed in $\Q_{ 131 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 87\cdot 131 + 44\cdot 131^{2} + 26\cdot 131^{3} + 31\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 19 + 95\cdot 131 + 73\cdot 131^{2} + 102\cdot 131^{3} + 92\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 22 + 74\cdot 131 + 38\cdot 131^{2} + 59\cdot 131^{3} + 16\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 36 + 6\cdot 131 + 8\cdot 131^{2} + 110\cdot 131^{3} + 108\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 40 + 84\cdot 131 + 31\cdot 131^{2} + 10\cdot 131^{3} + 83\cdot 131^{4} +O(131^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 62 + 117\cdot 131 + 125\cdot 131^{2} + 116\cdot 131^{3} + 13\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 87 + 9\cdot 131 + 103\cdot 131^{2} + 88\cdot 131^{3} + 39\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 125 + 49\cdot 131 + 98\cdot 131^{2} + 9\cdot 131^{3} + 7\cdot 131^{4} +O(131^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,6)(3,8)(4,5)$ | $-2$ |
| $4$ | $2$ | $(1,5)(2,8)(3,6)(4,7)$ | $0$ |
| $4$ | $2$ | $(1,3)(4,5)(7,8)$ | $0$ |
| $2$ | $4$ | $(1,3,7,8)(2,4,6,5)$ | $0$ |
| $2$ | $8$ | $(1,4,8,2,7,5,3,6)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,2,3,4,7,6,8,5)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.