Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.7247757312.3 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.24.2t1.b.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.0.3072.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 6x^{4} + 8x^{2} + 3 \)
|
The roots of $f$ are computed in $\Q_{ 131 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 4 + 25\cdot 131 + 102\cdot 131^{2} + 33\cdot 131^{3} + 127\cdot 131^{4} + 127\cdot 131^{5} +O(131^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 37 + 104\cdot 131 + 116\cdot 131^{2} + 93\cdot 131^{3} + 41\cdot 131^{4} + 72\cdot 131^{5} +O(131^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 50 + 51\cdot 131 + 22\cdot 131^{2} + 129\cdot 131^{3} + 55\cdot 131^{4} + 6\cdot 131^{5} +O(131^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 62 + 31\cdot 131 + 76\cdot 131^{2} + 100\cdot 131^{3} + 35\cdot 131^{4} + 63\cdot 131^{5} +O(131^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 69 + 99\cdot 131 + 54\cdot 131^{2} + 30\cdot 131^{3} + 95\cdot 131^{4} + 67\cdot 131^{5} +O(131^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 81 + 79\cdot 131 + 108\cdot 131^{2} + 131^{3} + 75\cdot 131^{4} + 124\cdot 131^{5} +O(131^{6})\)
|
| $r_{ 7 }$ | $=$ |
\( 94 + 26\cdot 131 + 14\cdot 131^{2} + 37\cdot 131^{3} + 89\cdot 131^{4} + 58\cdot 131^{5} +O(131^{6})\)
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| $r_{ 8 }$ | $=$ |
\( 127 + 105\cdot 131 + 28\cdot 131^{2} + 97\cdot 131^{3} + 3\cdot 131^{4} + 3\cdot 131^{5} +O(131^{6})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(2,7)(3,4)(5,6)$ | $0$ |
| $4$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ |
| $2$ | $8$ | $(1,3,2,5,8,6,7,4)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,5,7,3,8,4,2,6)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.