Basic invariants
| Dimension: | $2$ |
| Group: | $QD_{16}$ |
| Conductor: | \(2312\)\(\medspace = 2^{3} \cdot 17^{2} \) |
| Artin stem field: | Galois closure of 8.2.210093400576.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $QD_{16}$ |
| Parity: | odd |
| Determinant: | 1.136.2t1.b.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.2.39304.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 17x^{6} + 17x^{4} + 17x^{2} - 34 \)
|
The roots of $f$ are computed in $\Q_{ 179 }$ to precision 10.
Roots:
| $r_{ 1 }$ | $=$ |
\( 8 + 93\cdot 179 + 91\cdot 179^{2} + 158\cdot 179^{3} + 10\cdot 179^{4} + 129\cdot 179^{5} + 13\cdot 179^{6} + 17\cdot 179^{7} + 7\cdot 179^{8} + 153\cdot 179^{9} +O(179^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 27 + 18\cdot 179 + 8\cdot 179^{2} + 129\cdot 179^{4} + 36\cdot 179^{5} + 114\cdot 179^{6} + 167\cdot 179^{7} + 92\cdot 179^{8} + 117\cdot 179^{9} +O(179^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 36 + 156\cdot 179 + 146\cdot 179^{2} + 109\cdot 179^{3} + 152\cdot 179^{4} + 131\cdot 179^{5} + 176\cdot 179^{6} + 24\cdot 179^{7} + 52\cdot 179^{8} + 57\cdot 179^{9} +O(179^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 75 + 165\cdot 179 + 146\cdot 179^{2} + 91\cdot 179^{3} + 129\cdot 179^{4} + 11\cdot 179^{5} + 35\cdot 179^{6} + 150\cdot 179^{7} + 110\cdot 179^{8} + 10\cdot 179^{9} +O(179^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 104 + 13\cdot 179 + 32\cdot 179^{2} + 87\cdot 179^{3} + 49\cdot 179^{4} + 167\cdot 179^{5} + 143\cdot 179^{6} + 28\cdot 179^{7} + 68\cdot 179^{8} + 168\cdot 179^{9} +O(179^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 143 + 22\cdot 179 + 32\cdot 179^{2} + 69\cdot 179^{3} + 26\cdot 179^{4} + 47\cdot 179^{5} + 2\cdot 179^{6} + 154\cdot 179^{7} + 126\cdot 179^{8} + 121\cdot 179^{9} +O(179^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 152 + 160\cdot 179 + 170\cdot 179^{2} + 178\cdot 179^{3} + 49\cdot 179^{4} + 142\cdot 179^{5} + 64\cdot 179^{6} + 11\cdot 179^{7} + 86\cdot 179^{8} + 61\cdot 179^{9} +O(179^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 171 + 85\cdot 179 + 87\cdot 179^{2} + 20\cdot 179^{3} + 168\cdot 179^{4} + 49\cdot 179^{5} + 165\cdot 179^{6} + 161\cdot 179^{7} + 171\cdot 179^{8} + 25\cdot 179^{9} +O(179^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | |
| $4$ | $2$ | $(2,7)(3,4)(5,6)$ | $0$ | ✓ |
| $2$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $0$ | |
| $4$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $0$ | |
| $2$ | $8$ | $(1,3,2,4,8,6,7,5)$ | $\zeta_{8}^{3} + \zeta_{8}$ | |
| $2$ | $8$ | $(1,6,2,5,8,3,7,4)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |