Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8$ |
| Conductor: | \(278784\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 11^{2} \) |
| Frobenius-Schur indicator: | $-1$ |
| Root number: | $-1$ |
| Artin field: | Galois closure of 8.8.5416809268248576.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $Q_8$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{2}, \sqrt{33})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 132x^{6} + 2970x^{4} - 17424x^{2} + 17424 \)
|
The roots of $f$ are computed in $\Q_{ 167 }$ to precision 10.
Roots:
| $r_{ 1 }$ | $=$ |
\( 1 + 40\cdot 167 + 49\cdot 167^{2} + 38\cdot 167^{3} + 133\cdot 167^{4} + 124\cdot 167^{5} + 78\cdot 167^{6} + 37\cdot 167^{7} + 78\cdot 167^{8} + 18\cdot 167^{9} +O(167^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 14 + 14\cdot 167 + 96\cdot 167^{2} + 125\cdot 167^{3} + 72\cdot 167^{4} + 147\cdot 167^{5} + 97\cdot 167^{7} + 117\cdot 167^{8} + 94\cdot 167^{9} +O(167^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 53 + 95\cdot 167 + 74\cdot 167^{2} + 36\cdot 167^{3} + 92\cdot 167^{4} + 150\cdot 167^{5} + 134\cdot 167^{6} + 163\cdot 167^{7} + 68\cdot 167^{8} + 153\cdot 167^{9} +O(167^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 74 + 118\cdot 167 + 104\cdot 167^{2} + 16\cdot 167^{3} + 43\cdot 167^{4} + 94\cdot 167^{5} + 61\cdot 167^{6} + 139\cdot 167^{7} + 135\cdot 167^{8} + 3\cdot 167^{9} +O(167^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 93 + 48\cdot 167 + 62\cdot 167^{2} + 150\cdot 167^{3} + 123\cdot 167^{4} + 72\cdot 167^{5} + 105\cdot 167^{6} + 27\cdot 167^{7} + 31\cdot 167^{8} + 163\cdot 167^{9} +O(167^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 114 + 71\cdot 167 + 92\cdot 167^{2} + 130\cdot 167^{3} + 74\cdot 167^{4} + 16\cdot 167^{5} + 32\cdot 167^{6} + 3\cdot 167^{7} + 98\cdot 167^{8} + 13\cdot 167^{9} +O(167^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 153 + 152\cdot 167 + 70\cdot 167^{2} + 41\cdot 167^{3} + 94\cdot 167^{4} + 19\cdot 167^{5} + 166\cdot 167^{6} + 69\cdot 167^{7} + 49\cdot 167^{8} + 72\cdot 167^{9} +O(167^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 166 + 126\cdot 167 + 117\cdot 167^{2} + 128\cdot 167^{3} + 33\cdot 167^{4} + 42\cdot 167^{5} + 88\cdot 167^{6} + 129\cdot 167^{7} + 88\cdot 167^{8} + 148\cdot 167^{9} +O(167^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ | ✓ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ | |
| $2$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ | |
| $2$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ |