Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(3072\)\(\medspace = 2^{10} \cdot 3 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.57982058496.7 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.3.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.0.6144.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 24x^{4} - 32x^{2} + 24 \)
|
The roots of $f$ are computed in $\Q_{ 67 }$ to precision 7.
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 19\cdot 67 + 46\cdot 67^{2} + 64\cdot 67^{3} + 49\cdot 67^{4} + 21\cdot 67^{5} + 61\cdot 67^{6} +O(67^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 3 + 41\cdot 67 + 42\cdot 67^{2} + 3\cdot 67^{3} + 44\cdot 67^{4} + 46\cdot 67^{5} + 29\cdot 67^{6} +O(67^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 10 + 34\cdot 67 + 42\cdot 67^{2} + 2\cdot 67^{3} + 62\cdot 67^{4} + 11\cdot 67^{5} + 14\cdot 67^{6} +O(67^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 17 + 61\cdot 67 + 48\cdot 67^{3} + 31\cdot 67^{4} + 46\cdot 67^{5} + 15\cdot 67^{6} +O(67^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 50 + 5\cdot 67 + 66\cdot 67^{2} + 18\cdot 67^{3} + 35\cdot 67^{4} + 20\cdot 67^{5} + 51\cdot 67^{6} +O(67^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 57 + 32\cdot 67 + 24\cdot 67^{2} + 64\cdot 67^{3} + 4\cdot 67^{4} + 55\cdot 67^{5} + 52\cdot 67^{6} +O(67^{7})\)
|
| $r_{ 7 }$ | $=$ |
\( 64 + 25\cdot 67 + 24\cdot 67^{2} + 63\cdot 67^{3} + 22\cdot 67^{4} + 20\cdot 67^{5} + 37\cdot 67^{6} +O(67^{7})\)
|
| $r_{ 8 }$ | $=$ |
\( 65 + 47\cdot 67 + 20\cdot 67^{2} + 2\cdot 67^{3} + 17\cdot 67^{4} + 45\cdot 67^{5} + 5\cdot 67^{6} +O(67^{7})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(1,7)(2,8)(4,5)$ | $0$ |
| $4$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $0$ |
| $2$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ |
| $2$ | $8$ | $(1,3,7,5,8,6,2,4)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,5,2,3,8,4,7,6)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.