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 8.0.57982058496.8 |
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 $\Q_{ 283 }$ to precision 6.
Roots:
$r_{ 1 }$ | $=$ |
\( 34 + 219\cdot 283 + 78\cdot 283^{2} + 8\cdot 283^{3} + 118\cdot 283^{4} + 108\cdot 283^{5} +O(283^{6})\)
$r_{ 2 }$ |
$=$ |
\( 64 + 206\cdot 283 + 105\cdot 283^{2} + 262\cdot 283^{3} + 131\cdot 283^{4} + 197\cdot 283^{5} +O(283^{6})\)
| $r_{ 3 }$ |
$=$ |
\( 113 + 270\cdot 283 + 98\cdot 283^{2} + 224\cdot 283^{3} + 231\cdot 283^{4} + 123\cdot 283^{5} +O(283^{6})\)
| $r_{ 4 }$ |
$=$ |
\( 137 + 279\cdot 283 + 214\cdot 283^{2} + 66\cdot 283^{3} + 191\cdot 283^{4} + 210\cdot 283^{5} +O(283^{6})\)
| $r_{ 5 }$ |
$=$ |
\( 146 + 3\cdot 283 + 68\cdot 283^{2} + 216\cdot 283^{3} + 91\cdot 283^{4} + 72\cdot 283^{5} +O(283^{6})\)
| $r_{ 6 }$ |
$=$ |
\( 170 + 12\cdot 283 + 184\cdot 283^{2} + 58\cdot 283^{3} + 51\cdot 283^{4} + 159\cdot 283^{5} +O(283^{6})\)
| $r_{ 7 }$ |
$=$ |
\( 219 + 76\cdot 283 + 177\cdot 283^{2} + 20\cdot 283^{3} + 151\cdot 283^{4} + 85\cdot 283^{5} +O(283^{6})\)
| $r_{ 8 }$ |
$=$ |
\( 249 + 63\cdot 283 + 204\cdot 283^{2} + 274\cdot 283^{3} + 164\cdot 283^{4} + 174\cdot 283^{5} +O(283^{6})\)
| |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
$4$ | $2$ | $(2,5)(3,6)(4,7)$ | $0$ |
$4$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $0$ |
$2$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $0$ |
$2$ | $8$ | $(1,4,3,2,8,5,6,7)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
$2$ | $8$ | $(1,2,6,4,8,7,3,5)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.