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 8.8.343064484000000.1 |
| 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
| $f(x)$ | $=$ |
\( x^{8} - 105x^{6} + 3465x^{4} - 44100x^{2} + 176400 \)
|
The roots of $f$ are computed in $\Q_{ 109 }$ to precision 10.
Roots:
| $r_{ 1 }$ | $=$ |
\( 17 + 11\cdot 109 + 47\cdot 109^{2} + 81\cdot 109^{3} + 11\cdot 109^{4} + 96\cdot 109^{5} + 63\cdot 109^{6} + 91\cdot 109^{7} + 6\cdot 109^{8} + 87\cdot 109^{9} +O(109^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 27 + 17\cdot 109 + 67\cdot 109^{2} + 39\cdot 109^{3} + 40\cdot 109^{4} + 85\cdot 109^{5} + 9\cdot 109^{6} + 63\cdot 109^{7} + 99\cdot 109^{8} + 29\cdot 109^{9} +O(109^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 31 + 101\cdot 109 + 91\cdot 109^{2} + 8\cdot 109^{3} + 98\cdot 109^{4} + 35\cdot 109^{5} + 34\cdot 109^{6} + 91\cdot 109^{7} + 46\cdot 109^{8} + 52\cdot 109^{9} +O(109^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 52 + 15\cdot 109 + 61\cdot 109^{2} + 7\cdot 109^{3} + 29\cdot 109^{4} + 27\cdot 109^{5} + 18\cdot 109^{6} + 78\cdot 109^{7} + 19\cdot 109^{8} + 34\cdot 109^{9} +O(109^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 57 + 93\cdot 109 + 47\cdot 109^{2} + 101\cdot 109^{3} + 79\cdot 109^{4} + 81\cdot 109^{5} + 90\cdot 109^{6} + 30\cdot 109^{7} + 89\cdot 109^{8} + 74\cdot 109^{9} +O(109^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 78 + 7\cdot 109 + 17\cdot 109^{2} + 100\cdot 109^{3} + 10\cdot 109^{4} + 73\cdot 109^{5} + 74\cdot 109^{6} + 17\cdot 109^{7} + 62\cdot 109^{8} + 56\cdot 109^{9} +O(109^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 82 + 91\cdot 109 + 41\cdot 109^{2} + 69\cdot 109^{3} + 68\cdot 109^{4} + 23\cdot 109^{5} + 99\cdot 109^{6} + 45\cdot 109^{7} + 9\cdot 109^{8} + 79\cdot 109^{9} +O(109^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 92 + 97\cdot 109 + 61\cdot 109^{2} + 27\cdot 109^{3} + 97\cdot 109^{4} + 12\cdot 109^{5} + 45\cdot 109^{6} + 17\cdot 109^{7} + 102\cdot 109^{8} + 21\cdot 109^{9} +O(109^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ | ✓ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | |
| $2$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $0$ | |
| $2$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $0$ | |
| $2$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $0$ |