Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8$ |
| Conductor: | \(57600\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5^{2} \) |
| Frobenius-Schur indicator: | $-1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.0.47775744000000.2 |
| 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 \(\Q(\sqrt{5}, \sqrt{6})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 60x^{6} + 1170x^{4} + 9000x^{2} + 22500 \)
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The roots of $f$ are computed in $\Q_{ 29 }$ to precision 10.
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 20\cdot 29 + 10\cdot 29^{2} + 10\cdot 29^{3} + 9\cdot 29^{4} + 7\cdot 29^{5} + 12\cdot 29^{6} + 15\cdot 29^{7} + 5\cdot 29^{8} + 21\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 4 + 16\cdot 29 + 21\cdot 29^{2} + 18\cdot 29^{3} + 28\cdot 29^{4} + 18\cdot 29^{5} + 7\cdot 29^{6} + 27\cdot 29^{7} + 10\cdot 29^{8} + 19\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 9 + 9\cdot 29 + 29^{2} + 2\cdot 29^{3} + 19\cdot 29^{5} + 29^{6} + 7\cdot 29^{7} + 8\cdot 29^{8} + 27\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 10 + 26\cdot 29 + 18\cdot 29^{2} + 6\cdot 29^{3} + 27\cdot 29^{4} + 15\cdot 29^{5} + 14\cdot 29^{6} + 7\cdot 29^{7} + 25\cdot 29^{8} + 16\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 19 + 2\cdot 29 + 10\cdot 29^{2} + 22\cdot 29^{3} + 29^{4} + 13\cdot 29^{5} + 14\cdot 29^{6} + 21\cdot 29^{7} + 3\cdot 29^{8} + 12\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 20 + 19\cdot 29 + 27\cdot 29^{2} + 26\cdot 29^{3} + 28\cdot 29^{4} + 9\cdot 29^{5} + 27\cdot 29^{6} + 21\cdot 29^{7} + 20\cdot 29^{8} + 29^{9} +O(29^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 25 + 12\cdot 29 + 7\cdot 29^{2} + 10\cdot 29^{3} + 10\cdot 29^{5} + 21\cdot 29^{6} + 29^{7} + 18\cdot 29^{8} + 9\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 27 + 8\cdot 29 + 18\cdot 29^{2} + 18\cdot 29^{3} + 19\cdot 29^{4} + 21\cdot 29^{5} + 16\cdot 29^{6} + 13\cdot 29^{7} + 23\cdot 29^{8} + 7\cdot 29^{9} +O(29^{10})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $0$ |
| $2$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ |
| $2$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.