Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8$ |
| Conductor: | \(112896\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 7^{2} \) |
| Frobenius-Schur indicator: | $-1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.8.1438916737499136.2 |
| 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 \(\Q(\sqrt{3}, \sqrt{14})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 84x^{6} + 1260x^{4} - 5292x^{2} + 441 \)
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The roots of $f$ are computed in $\Q_{ 47 }$ to precision 10.
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 28\cdot 47 + 45\cdot 47^{2} + 33\cdot 47^{3} + 26\cdot 47^{4} + 31\cdot 47^{5} + 18\cdot 47^{6} + 35\cdot 47^{7} + 17\cdot 47^{8} + 29\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 9 + 36\cdot 47 + 20\cdot 47^{2} + 8\cdot 47^{3} + 8\cdot 47^{4} + 44\cdot 47^{5} + 19\cdot 47^{6} + 27\cdot 47^{7} + 4\cdot 47^{8} + 35\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 13 + 25\cdot 47 + 39\cdot 47^{2} + 45\cdot 47^{3} + 26\cdot 47^{4} + 6\cdot 47^{5} + 7\cdot 47^{6} + 16\cdot 47^{7} + 44\cdot 47^{8} + 3\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 21 + 12\cdot 47 + 24\cdot 47^{2} + 21\cdot 47^{3} + 29\cdot 47^{4} + 29\cdot 47^{5} + 29\cdot 47^{6} + 3\cdot 47^{7} + 22\cdot 47^{8} + 44\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 26 + 34\cdot 47 + 22\cdot 47^{2} + 25\cdot 47^{3} + 17\cdot 47^{4} + 17\cdot 47^{5} + 17\cdot 47^{6} + 43\cdot 47^{7} + 24\cdot 47^{8} + 2\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 34 + 21\cdot 47 + 7\cdot 47^{2} + 47^{3} + 20\cdot 47^{4} + 40\cdot 47^{5} + 39\cdot 47^{6} + 30\cdot 47^{7} + 2\cdot 47^{8} + 43\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 38 + 10\cdot 47 + 26\cdot 47^{2} + 38\cdot 47^{3} + 38\cdot 47^{4} + 2\cdot 47^{5} + 27\cdot 47^{6} + 19\cdot 47^{7} + 42\cdot 47^{8} + 11\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 45 + 18\cdot 47 + 47^{2} + 13\cdot 47^{3} + 20\cdot 47^{4} + 15\cdot 47^{5} + 28\cdot 47^{6} + 11\cdot 47^{7} + 29\cdot 47^{8} + 17\cdot 47^{9} +O(47^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ | ✓ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ | |
| $2$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ | |
| $2$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ |