Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(3328\)\(\medspace = 2^{8} \cdot 13 \) |
| Artin stem field: | Galois closure of 8.0.2835349504.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.104.2t1.b.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(i, \sqrt{13})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 24x^{4} + 169 \)
|
The roots of $f$ are computed in $\Q_{ 257 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 21 + 242\cdot 257 + 90\cdot 257^{2} + 145\cdot 257^{3} + 234\cdot 257^{4} +O(257^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 79 + 186\cdot 257 + 176\cdot 257^{2} + 241\cdot 257^{3} + 113\cdot 257^{4} +O(257^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 87 + 24\cdot 257 + 118\cdot 257^{2} + 21\cdot 257^{3} + 10\cdot 257^{4} +O(257^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 107 + 57\cdot 257 + 36\cdot 257^{2} + 70\cdot 257^{3} + 98\cdot 257^{4} +O(257^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 150 + 199\cdot 257 + 220\cdot 257^{2} + 186\cdot 257^{3} + 158\cdot 257^{4} +O(257^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 170 + 232\cdot 257 + 138\cdot 257^{2} + 235\cdot 257^{3} + 246\cdot 257^{4} +O(257^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 178 + 70\cdot 257 + 80\cdot 257^{2} + 15\cdot 257^{3} + 143\cdot 257^{4} +O(257^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 236 + 14\cdot 257 + 166\cdot 257^{2} + 111\cdot 257^{3} + 22\cdot 257^{4} +O(257^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | |
| $2$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $0$ | ✓ |
| $2$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ | |
| $2$ | $2$ | $(3,6)(4,5)$ | $0$ | |
| $1$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $2 \zeta_{4}$ | |
| $1$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $-2 \zeta_{4}$ | |
| $2$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $0$ | |
| $2$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $0$ | |
| $2$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $0$ |