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 8.8.7644119040000.1 |
| 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 \(\Q(\sqrt{2}, \sqrt{3})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 60x^{6} + 900x^{4} - 4500x^{2} + 5625 \)
|
The roots of $f$ are computed in $\Q_{ 47 }$ to precision 10.
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 43\cdot 47 + 29\cdot 47^{2} + 8\cdot 47^{3} + 3\cdot 47^{4} + 8\cdot 47^{5} + 44\cdot 47^{6} + 17\cdot 47^{7} + 5\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 18 + 6\cdot 47 + 34\cdot 47^{2} + 17\cdot 47^{3} + 32\cdot 47^{4} + 43\cdot 47^{5} + 23\cdot 47^{6} + 33\cdot 47^{7} + 31\cdot 47^{8} + 17\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 19 + 23\cdot 47 + 21\cdot 47^{2} + 33\cdot 47^{3} + 16\cdot 47^{5} + 36\cdot 47^{6} + 36\cdot 47^{7} + 32\cdot 47^{8} + 13\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 20 + 47 + 5\cdot 47^{2} + 23\cdot 47^{3} + 11\cdot 47^{4} + 6\cdot 47^{5} + 14\cdot 47^{6} + 18\cdot 47^{7} + 23\cdot 47^{8} + 7\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 27 + 45\cdot 47 + 41\cdot 47^{2} + 23\cdot 47^{3} + 35\cdot 47^{4} + 40\cdot 47^{5} + 32\cdot 47^{6} + 28\cdot 47^{7} + 23\cdot 47^{8} + 39\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 28 + 23\cdot 47 + 25\cdot 47^{2} + 13\cdot 47^{3} + 46\cdot 47^{4} + 30\cdot 47^{5} + 10\cdot 47^{6} + 10\cdot 47^{7} + 14\cdot 47^{8} + 33\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 29 + 40\cdot 47 + 12\cdot 47^{2} + 29\cdot 47^{3} + 14\cdot 47^{4} + 3\cdot 47^{5} + 23\cdot 47^{6} + 13\cdot 47^{7} + 15\cdot 47^{8} + 29\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 44 + 3\cdot 47 + 17\cdot 47^{2} + 38\cdot 47^{3} + 43\cdot 47^{4} + 38\cdot 47^{5} + 2\cdot 47^{6} + 29\cdot 47^{7} + 46\cdot 47^{8} + 41\cdot 47^{9} +O(47^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ | ✓ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | |
| $2$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ | |
| $2$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $0$ | |
| $2$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $0$ |