Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8$ |
| Conductor: | \(176400\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \) |
| Frobenius-Schur indicator: | $-1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.0.343064484000000.1 |
| 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 \(\Q(\sqrt{5}, \sqrt{21})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 105x^{6} + 3780x^{4} + 55125x^{2} + 275625 \)
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The roots of $f$ are computed in $\Q_{ 41 }$ to precision 10.
Roots:
| $r_{ 1 }$ | $=$ |
\( 5 + 21\cdot 41 + 10\cdot 41^{2} + 24\cdot 41^{3} + 13\cdot 41^{4} + 27\cdot 41^{5} + 19\cdot 41^{6} + 10\cdot 41^{7} + 8\cdot 41^{8} + 12\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 6 + 29\cdot 41 + 22\cdot 41^{2} + 8\cdot 41^{3} + 9\cdot 41^{4} + 5\cdot 41^{5} + 6\cdot 41^{6} + 32\cdot 41^{7} + 33\cdot 41^{8} + 22\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 10 + 31\cdot 41 + 10\cdot 41^{2} + 31\cdot 41^{3} + 16\cdot 41^{4} + 38\cdot 41^{5} + 19\cdot 41^{6} + 5\cdot 41^{7} + 4\cdot 41^{8} + 10\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 12 + 12\cdot 41 + 31\cdot 41^{2} + 39\cdot 41^{3} + 37\cdot 41^{4} + 25\cdot 41^{5} + 11\cdot 41^{6} + 16\cdot 41^{7} + 14\cdot 41^{8} + 23\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 29 + 28\cdot 41 + 9\cdot 41^{2} + 41^{3} + 3\cdot 41^{4} + 15\cdot 41^{5} + 29\cdot 41^{6} + 24\cdot 41^{7} + 26\cdot 41^{8} + 17\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 31 + 9\cdot 41 + 30\cdot 41^{2} + 9\cdot 41^{3} + 24\cdot 41^{4} + 2\cdot 41^{5} + 21\cdot 41^{6} + 35\cdot 41^{7} + 36\cdot 41^{8} + 30\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 35 + 11\cdot 41 + 18\cdot 41^{2} + 32\cdot 41^{3} + 31\cdot 41^{4} + 35\cdot 41^{5} + 34\cdot 41^{6} + 8\cdot 41^{7} + 7\cdot 41^{8} + 18\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 36 + 19\cdot 41 + 30\cdot 41^{2} + 16\cdot 41^{3} + 27\cdot 41^{4} + 13\cdot 41^{5} + 21\cdot 41^{6} + 30\cdot 41^{7} + 32\cdot 41^{8} + 28\cdot 41^{9} +O(41^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ |
| $2$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ |
| $2$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.