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 8.0.47775744000000.1 |
| 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 \(\Q(\sqrt{5}, \sqrt{6})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 60x^{6} + 810x^{4} + 1800x^{2} + 900 \)
|
The roots of $f$ are computed in $\Q_{ 71 }$ to precision 10.
Roots:
| $r_{ 1 }$ | $=$ |
\( 12 + 34\cdot 71 + 41\cdot 71^{2} + 46\cdot 71^{3} + 49\cdot 71^{4} + 46\cdot 71^{5} + 71^{6} + 4\cdot 71^{7} + 12\cdot 71^{8} + 46\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 21 + 11\cdot 71 + 49\cdot 71^{2} + 64\cdot 71^{3} + 60\cdot 71^{4} + 23\cdot 71^{5} + 9\cdot 71^{6} + 58\cdot 71^{7} + 45\cdot 71^{8} + 52\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 24 + 12\cdot 71 + 67\cdot 71^{2} + 24\cdot 71^{3} + 26\cdot 71^{4} + 25\cdot 71^{5} + 53\cdot 71^{6} + 60\cdot 71^{7} + 46\cdot 71^{8} + 48\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 25 + 5\cdot 71 + 64\cdot 71^{2} + 20\cdot 71^{3} + 28\cdot 71^{4} + 27\cdot 71^{5} + 35\cdot 71^{6} + 11\cdot 71^{7} + 32\cdot 71^{8} + 50\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 46 + 65\cdot 71 + 6\cdot 71^{2} + 50\cdot 71^{3} + 42\cdot 71^{4} + 43\cdot 71^{5} + 35\cdot 71^{6} + 59\cdot 71^{7} + 38\cdot 71^{8} + 20\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 47 + 58\cdot 71 + 3\cdot 71^{2} + 46\cdot 71^{3} + 44\cdot 71^{4} + 45\cdot 71^{5} + 17\cdot 71^{6} + 10\cdot 71^{7} + 24\cdot 71^{8} + 22\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 50 + 59\cdot 71 + 21\cdot 71^{2} + 6\cdot 71^{3} + 10\cdot 71^{4} + 47\cdot 71^{5} + 61\cdot 71^{6} + 12\cdot 71^{7} + 25\cdot 71^{8} + 18\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 59 + 36\cdot 71 + 29\cdot 71^{2} + 24\cdot 71^{3} + 21\cdot 71^{4} + 24\cdot 71^{5} + 69\cdot 71^{6} + 66\cdot 71^{7} + 58\cdot 71^{8} + 24\cdot 71^{9} +O(71^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | ✓ |
| $2$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ | |
| $2$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $0$ | |
| $2$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $0$ |