Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(164\)\(\medspace = 2^{2} \cdot 41 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 8.0.17643776.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Projective image: | $D_4$ |
| Projective field: | Galois closure of 4.0.656.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 173 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 19 + 129\cdot 173 + 45\cdot 173^{2} + 42\cdot 173^{3} + 6\cdot 173^{4} +O(173^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 22 + 34\cdot 173 + 73\cdot 173^{2} + 24\cdot 173^{3} + 138\cdot 173^{4} +O(173^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 26 + 157\cdot 173 + 26\cdot 173^{2} + 149\cdot 173^{3} + 42\cdot 173^{4} +O(173^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 75 + 68\cdot 173 + 129\cdot 173^{2} + 126\cdot 173^{3} + 84\cdot 173^{4} +O(173^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 112 + 61\cdot 173 + 20\cdot 173^{2} + 87\cdot 173^{3} + 68\cdot 173^{4} +O(173^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 130 + 9\cdot 173 + 21\cdot 173^{2} + 75\cdot 173^{3} + 148\cdot 173^{4} +O(173^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 149 + 149\cdot 173 + 165\cdot 173^{2} + 131\cdot 173^{3} + 90\cdot 173^{4} +O(173^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 159 + 81\cdot 173 + 36\cdot 173^{2} + 55\cdot 173^{3} + 112\cdot 173^{4} +O(173^{5})\)
|
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 values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,8)(2,4)(3,6)(5,7)$ | $-2$ | $-2$ |
| $4$ | $2$ | $(1,4)(2,8)(3,5)(6,7)$ | $0$ | $0$ |
| $4$ | $2$ | $(1,7)(3,6)(5,8)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,7,8,5)(2,6,4,3)$ | $0$ | $0$ |
| $2$ | $8$ | $(1,4,7,3,8,2,5,6)$ | $-\zeta_{8}^{3} + \zeta_{8}$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,3,5,4,8,6,7,2)$ | $\zeta_{8}^{3} - \zeta_{8}$ | $-\zeta_{8}^{3} + \zeta_{8}$ |