Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(495\)\(\medspace = 3^{2} \cdot 5 \cdot 11 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.606436875.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.55.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.2.2475.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + x^{6} - 3x^{5} + 9x^{4} + 6x^{3} + 16x^{2} - 5 \)
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The roots of $f$ are computed in $\Q_{ 179 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 5 + 117\cdot 179 + 22\cdot 179^{2} + 40\cdot 179^{3} + 18\cdot 179^{4} +O(179^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 11 + 173\cdot 179 + 61\cdot 179^{2} + 18\cdot 179^{3} + 145\cdot 179^{4} +O(179^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 37 + 100\cdot 179 + 100\cdot 179^{2} + 28\cdot 179^{3} + 144\cdot 179^{4} +O(179^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 49 + 85\cdot 179 + 147\cdot 179^{2} + 94\cdot 179^{3} + 17\cdot 179^{4} +O(179^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 58 + 70\cdot 179 + 31\cdot 179^{2} + 47\cdot 179^{3} + 55\cdot 179^{4} +O(179^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 87 + 134\cdot 179 + 169\cdot 179^{2} + 72\cdot 179^{3} + 68\cdot 179^{4} +O(179^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 135 + 58\cdot 179 + 43\cdot 179^{2} + 131\cdot 179^{3} + 52\cdot 179^{4} +O(179^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 155 + 155\cdot 179 + 138\cdot 179^{2} + 103\cdot 179^{3} + 35\cdot 179^{4} +O(179^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,6)(3,4)(7,8)$ | $-2$ |
| $4$ | $2$ | $(1,8)(2,4)(3,6)(5,7)$ | $0$ |
| $4$ | $2$ | $(1,5)(2,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,3,5,4)(2,7,6,8)$ | $0$ |
| $2$ | $8$ | $(1,6,3,8,5,2,4,7)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,8,4,6,5,7,3,2)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.