Basic invariants
Dimension: | $2$ |
Group: | $D_{8}$ |
Conductor: | \(1280\)\(\medspace = 2^{8} \cdot 5 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 8.0.2097152000.6 |
Galois orbit size: | $2$ |
Smallest permutation container: | $D_{8}$ |
Parity: | odd |
Projective image: | $D_4$ |
Projective field: | Galois closure of 4.0.1280.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 109 }$ to precision 6.
Roots:
$r_{ 1 }$ | $=$ | \( 18 + 45\cdot 109 + 81\cdot 109^{2} + 92\cdot 109^{3} + 27\cdot 109^{5} +O(109^{6})\) |
$r_{ 2 }$ | $=$ | \( 22 + 10\cdot 109 + 12\cdot 109^{2} + 9\cdot 109^{3} + 70\cdot 109^{4} + 50\cdot 109^{5} +O(109^{6})\) |
$r_{ 3 }$ | $=$ | \( 26 + 99\cdot 109 + 5\cdot 109^{2} + 99\cdot 109^{3} + 12\cdot 109^{4} + 102\cdot 109^{5} +O(109^{6})\) |
$r_{ 4 }$ | $=$ | \( 41 + 46\cdot 109 + 7\cdot 109^{2} + 56\cdot 109^{3} + 84\cdot 109^{4} + 79\cdot 109^{5} +O(109^{6})\) |
$r_{ 5 }$ | $=$ | \( 68 + 62\cdot 109 + 101\cdot 109^{2} + 52\cdot 109^{3} + 24\cdot 109^{4} + 29\cdot 109^{5} +O(109^{6})\) |
$r_{ 6 }$ | $=$ | \( 83 + 9\cdot 109 + 103\cdot 109^{2} + 9\cdot 109^{3} + 96\cdot 109^{4} + 6\cdot 109^{5} +O(109^{6})\) |
$r_{ 7 }$ | $=$ | \( 87 + 98\cdot 109 + 96\cdot 109^{2} + 99\cdot 109^{3} + 38\cdot 109^{4} + 58\cdot 109^{5} +O(109^{6})\) |
$r_{ 8 }$ | $=$ | \( 91 + 63\cdot 109 + 27\cdot 109^{2} + 16\cdot 109^{3} + 108\cdot 109^{4} + 81\cdot 109^{5} +O(109^{6})\) |
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 values | |
$c1$ | $c2$ | |||
$1$ | $1$ | $()$ | $2$ | $2$ |
$1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | $-2$ |
$4$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ | $0$ |
$4$ | $2$ | $(1,7)(2,8)(3,6)$ | $0$ | $0$ |
$2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ | $0$ |
$2$ | $8$ | $(1,6,2,4,8,3,7,5)$ | $-\zeta_{8}^{3} + \zeta_{8}$ | $\zeta_{8}^{3} - \zeta_{8}$ |
$2$ | $8$ | $(1,4,7,6,8,5,2,3)$ | $\zeta_{8}^{3} - \zeta_{8}$ | $-\zeta_{8}^{3} + \zeta_{8}$ |