Basic invariants
Dimension: | $2$ |
Group: | $D_{8}$ |
Conductor: | \(1088\)\(\medspace = 2^{6} \cdot 17 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 8.0.321978368.4 |
Galois orbit size: | $2$ |
Smallest permutation container: | $D_{8}$ |
Parity: | odd |
Determinant: | 1.68.2t1.a.a |
Projective image: | $D_4$ |
Projective stem field: | Galois closure of 4.0.272.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 4x^{6} + 11x^{4} - 20x^{2} + 17 \) . |
The roots of $f$ are computed in $\Q_{ 53 }$ to precision 6.
Roots:
$r_{ 1 }$ | $=$ |
\( 9 + 29\cdot 53 + 16\cdot 53^{2} + 17\cdot 53^{3} + 42\cdot 53^{4} + 31\cdot 53^{5} +O(53^{6})\)
$r_{ 2 }$ |
$=$ |
\( 19 + 37\cdot 53 + 19\cdot 53^{2} + 29\cdot 53^{3} + 36\cdot 53^{4} + 15\cdot 53^{5} +O(53^{6})\)
| $r_{ 3 }$ |
$=$ |
\( 23 + 12\cdot 53 + 18\cdot 53^{2} + 44\cdot 53^{3} + 52\cdot 53^{4} + 14\cdot 53^{5} +O(53^{6})\)
| $r_{ 4 }$ |
$=$ |
\( 26 + 20\cdot 53 + 48\cdot 53^{2} + 38\cdot 53^{3} + 41\cdot 53^{4} + 29\cdot 53^{5} +O(53^{6})\)
| $r_{ 5 }$ |
$=$ |
\( 27 + 32\cdot 53 + 4\cdot 53^{2} + 14\cdot 53^{3} + 11\cdot 53^{4} + 23\cdot 53^{5} +O(53^{6})\)
| $r_{ 6 }$ |
$=$ |
\( 30 + 40\cdot 53 + 34\cdot 53^{2} + 8\cdot 53^{3} + 38\cdot 53^{5} +O(53^{6})\)
| $r_{ 7 }$ |
$=$ |
\( 34 + 15\cdot 53 + 33\cdot 53^{2} + 23\cdot 53^{3} + 16\cdot 53^{4} + 37\cdot 53^{5} +O(53^{6})\)
| $r_{ 8 }$ |
$=$ |
\( 44 + 23\cdot 53 + 36\cdot 53^{2} + 35\cdot 53^{3} + 10\cdot 53^{4} + 21\cdot 53^{5} +O(53^{6})\)
| |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
$4$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $0$ |
$4$ | $2$ | $(1,2)(4,5)(7,8)$ | $0$ |
$2$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ |
$2$ | $8$ | $(1,3,2,4,8,6,7,5)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
$2$ | $8$ | $(1,4,7,3,8,5,2,6)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.