Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(1024\)\(\medspace = 2^{10} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.2147483648.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.4.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.2.2048.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 8x^{4} - 2 \)
|
The roots of $f$ are computed in $\Q_{ 257 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 32\cdot 257 + 120\cdot 257^{2} + 169\cdot 257^{3} + 247\cdot 257^{4} +O(257^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 39 + 180\cdot 257 + 227\cdot 257^{2} + 217\cdot 257^{3} + 16\cdot 257^{4} +O(257^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 48 + 22\cdot 257 + 140\cdot 257^{2} + 24\cdot 257^{3} + 12\cdot 257^{4} +O(257^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 110 + 110\cdot 257 + 178\cdot 257^{2} + 158\cdot 257^{3} + 177\cdot 257^{4} +O(257^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 147 + 146\cdot 257 + 78\cdot 257^{2} + 98\cdot 257^{3} + 79\cdot 257^{4} +O(257^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 209 + 234\cdot 257 + 116\cdot 257^{2} + 232\cdot 257^{3} + 244\cdot 257^{4} +O(257^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 218 + 76\cdot 257 + 29\cdot 257^{2} + 39\cdot 257^{3} + 240\cdot 257^{4} +O(257^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 254 + 224\cdot 257 + 136\cdot 257^{2} + 87\cdot 257^{3} + 9\cdot 257^{4} +O(257^{5})\)
|
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 | Complex conjugation |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | |
| $4$ | $2$ | $(1,8)(2,4)(5,7)$ | $0$ | ✓ |
| $4$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ | |
| $2$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $0$ | |
| $2$ | $8$ | $(1,7,3,5,8,2,6,4)$ | $\zeta_{8}^{3} - \zeta_{8}$ | |
| $2$ | $8$ | $(1,5,6,7,8,4,3,2)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |