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 8.0.5416809268248576.1 |
| 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
| $f(x)$ | $=$ |
\( x^{8} + 132x^{6} + 2970x^{4} + 17424x^{2} + 17424 \)
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The roots of $f$ are computed in $\Q_{ 31 }$ to precision 10.
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 23\cdot 31 + 11\cdot 31^{2} + 27\cdot 31^{4} + 14\cdot 31^{5} + 27\cdot 31^{6} + 30\cdot 31^{7} + 11\cdot 31^{8} + 15\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 8 + 30\cdot 31 + 4\cdot 31^{2} + 13\cdot 31^{3} + 12\cdot 31^{4} + 16\cdot 31^{5} + 6\cdot 31^{6} + 29\cdot 31^{7} + 5\cdot 31^{8} + 21\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 10 + 23\cdot 31 + 9\cdot 31^{2} + 14\cdot 31^{3} + 31^{5} + 2\cdot 31^{6} + 22\cdot 31^{7} + 28\cdot 31^{8} + 21\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 14 + 29\cdot 31 + 26\cdot 31^{2} + 19\cdot 31^{3} + 10\cdot 31^{4} + 14\cdot 31^{5} + 7\cdot 31^{6} + 25\cdot 31^{7} + 28\cdot 31^{8} + 27\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 17 + 31 + 4\cdot 31^{2} + 11\cdot 31^{3} + 20\cdot 31^{4} + 16\cdot 31^{5} + 23\cdot 31^{6} + 5\cdot 31^{7} + 2\cdot 31^{8} + 3\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 21 + 7\cdot 31 + 21\cdot 31^{2} + 16\cdot 31^{3} + 30\cdot 31^{4} + 29\cdot 31^{5} + 28\cdot 31^{6} + 8\cdot 31^{7} + 2\cdot 31^{8} + 9\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 23 + 26\cdot 31^{2} + 17\cdot 31^{3} + 18\cdot 31^{4} + 14\cdot 31^{5} + 24\cdot 31^{6} + 31^{7} + 25\cdot 31^{8} + 9\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 29 + 7\cdot 31 + 19\cdot 31^{2} + 30\cdot 31^{3} + 3\cdot 31^{4} + 16\cdot 31^{5} + 3\cdot 31^{6} + 19\cdot 31^{8} + 15\cdot 31^{9} +O(31^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ |
| $2$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $0$ |
| $2$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.