Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(2624\)\(\medspace = 2^{6} \cdot 41 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.4516806656.3 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.164.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.0.656.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 8x^{6} + 42x^{4} - 72x^{2} + 41 \)
|
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 8.
Roots:
| $r_{ 1 }$ | $=$ |
\( 5 + 3\cdot 61 + 23\cdot 61^{2} + 27\cdot 61^{3} + 36\cdot 61^{4} + 26\cdot 61^{5} + 53\cdot 61^{6} + 53\cdot 61^{7} +O(61^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 20 + 17\cdot 61 + 33\cdot 61^{2} + 32\cdot 61^{3} + 29\cdot 61^{4} + 5\cdot 61^{5} + 32\cdot 61^{6} + 19\cdot 61^{7} +O(61^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 21 + 47\cdot 61 + 36\cdot 61^{2} + 24\cdot 61^{3} + 26\cdot 61^{4} + 35\cdot 61^{5} + 7\cdot 61^{6} + 45\cdot 61^{7} +O(61^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 22 + 30\cdot 61 + 22\cdot 61^{2} + 60\cdot 61^{3} + 54\cdot 61^{4} + 55\cdot 61^{5} + 5\cdot 61^{6} + 22\cdot 61^{7} +O(61^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 39 + 30\cdot 61 + 38\cdot 61^{2} + 6\cdot 61^{4} + 5\cdot 61^{5} + 55\cdot 61^{6} + 38\cdot 61^{7} +O(61^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( 40 + 13\cdot 61 + 24\cdot 61^{2} + 36\cdot 61^{3} + 34\cdot 61^{4} + 25\cdot 61^{5} + 53\cdot 61^{6} + 15\cdot 61^{7} +O(61^{8})\)
|
| $r_{ 7 }$ | $=$ |
\( 41 + 43\cdot 61 + 27\cdot 61^{2} + 28\cdot 61^{3} + 31\cdot 61^{4} + 55\cdot 61^{5} + 28\cdot 61^{6} + 41\cdot 61^{7} +O(61^{8})\)
|
| $r_{ 8 }$ | $=$ |
\( 56 + 57\cdot 61 + 37\cdot 61^{2} + 33\cdot 61^{3} + 24\cdot 61^{4} + 34\cdot 61^{5} + 7\cdot 61^{6} + 7\cdot 61^{7} +O(61^{8})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(1,7)(2,8)(3,5)(4,6)$ | $0$ |
| $4$ | $2$ | $(2,6)(3,7)(4,5)$ | $0$ |
| $2$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ |
| $2$ | $8$ | $(1,3,4,2,8,6,5,7)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,2,5,3,8,7,4,6)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.