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 8.8.5416809268248576.2 |
| 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 \(\Q(\sqrt{2}, \sqrt{33})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 132x^{6} + 5346x^{4} - 69696x^{2} + 278784 \)
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The roots of $f$ are computed in $\Q_{ 41 }$ to precision 10.
Roots:
| $r_{ 1 }$ | $=$ |
\( 10 + 25\cdot 41 + 41^{2} + 30\cdot 41^{3} + 11\cdot 41^{4} + 34\cdot 41^{5} + 40\cdot 41^{6} + 20\cdot 41^{7} + 26\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 15 + 13\cdot 41 + 4\cdot 41^{2} + 35\cdot 41^{3} + 35\cdot 41^{4} + 35\cdot 41^{5} + 31\cdot 41^{6} + 21\cdot 41^{7} + 21\cdot 41^{8} + 10\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 16 + 13\cdot 41 + 10\cdot 41^{2} + 15\cdot 41^{3} + 22\cdot 41^{4} + 9\cdot 41^{5} + 6\cdot 41^{6} + 25\cdot 41^{7} + 5\cdot 41^{8} + 2\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 17 + 31\cdot 41 + 15\cdot 41^{2} + 38\cdot 41^{3} + 30\cdot 41^{4} + 4\cdot 41^{5} + 39\cdot 41^{6} + 2\cdot 41^{7} + 17\cdot 41^{8} + 36\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 24 + 9\cdot 41 + 25\cdot 41^{2} + 2\cdot 41^{3} + 10\cdot 41^{4} + 36\cdot 41^{5} + 41^{6} + 38\cdot 41^{7} + 23\cdot 41^{8} + 4\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 25 + 27\cdot 41 + 30\cdot 41^{2} + 25\cdot 41^{3} + 18\cdot 41^{4} + 31\cdot 41^{5} + 34\cdot 41^{6} + 15\cdot 41^{7} + 35\cdot 41^{8} + 38\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 26 + 27\cdot 41 + 36\cdot 41^{2} + 5\cdot 41^{3} + 5\cdot 41^{4} + 5\cdot 41^{5} + 9\cdot 41^{6} + 19\cdot 41^{7} + 19\cdot 41^{8} + 30\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 31 + 15\cdot 41 + 39\cdot 41^{2} + 10\cdot 41^{3} + 29\cdot 41^{4} + 6\cdot 41^{5} + 20\cdot 41^{7} + 40\cdot 41^{8} + 14\cdot 41^{9} +O(41^{10})\)
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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 |
| $1$ | $1$ | $()$ | $2$ | ✓ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ | |
| $2$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ | |
| $2$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ |