Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8$ |
| Conductor: | \(92416\)\(\medspace = 2^{8} \cdot 19^{2} \) |
| Frobenius-Schur indicator: | $-1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.8.789298907447296.1 |
| 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 \(\Q(\sqrt{2}, \sqrt{19})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 76x^{6} + 1748x^{4} - 12996x^{2} + 29241 \)
|
The roots of $f$ are computed in $\Q_{ 31 }$ to precision 10.
Roots:
| $r_{ 1 }$ | $=$ |
\( 1 + 30\cdot 31 + 2\cdot 31^{2} + 24\cdot 31^{3} + 3\cdot 31^{4} + 13\cdot 31^{5} + 2\cdot 31^{6} + 10\cdot 31^{7} + 7\cdot 31^{8} + 18\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 3 + 2\cdot 31 + 20\cdot 31^{2} + 15\cdot 31^{3} + 24\cdot 31^{4} + 17\cdot 31^{5} + 26\cdot 31^{6} + 19\cdot 31^{7} + 26\cdot 31^{8} + 9\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 4 + 24\cdot 31 + 4\cdot 31^{2} + 18\cdot 31^{3} + 26\cdot 31^{4} + 22\cdot 31^{5} + 13\cdot 31^{6} + 9\cdot 31^{7} + 6\cdot 31^{8} + 4\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 9 + 18\cdot 31 + 2\cdot 31^{2} + 29\cdot 31^{3} + 18\cdot 31^{4} + 13\cdot 31^{5} + 21\cdot 31^{6} + 16\cdot 31^{7} + 29\cdot 31^{8} + 7\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 22 + 12\cdot 31 + 28\cdot 31^{2} + 31^{3} + 12\cdot 31^{4} + 17\cdot 31^{5} + 9\cdot 31^{6} + 14\cdot 31^{7} + 31^{8} + 23\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 27 + 6\cdot 31 + 26\cdot 31^{2} + 12\cdot 31^{3} + 4\cdot 31^{4} + 8\cdot 31^{5} + 17\cdot 31^{6} + 21\cdot 31^{7} + 24\cdot 31^{8} + 26\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 28 + 28\cdot 31 + 10\cdot 31^{2} + 15\cdot 31^{3} + 6\cdot 31^{4} + 13\cdot 31^{5} + 4\cdot 31^{6} + 11\cdot 31^{7} + 4\cdot 31^{8} + 21\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 30 + 28\cdot 31^{2} + 6\cdot 31^{3} + 27\cdot 31^{4} + 17\cdot 31^{5} + 28\cdot 31^{6} + 20\cdot 31^{7} + 23\cdot 31^{8} + 12\cdot 31^{9} +O(31^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ | ✓ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | |
| $2$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $0$ | |
| $2$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $0$ | |
| $2$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ |