Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8$ |
| Conductor: | \(112896\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 7^{2} \) |
| Frobenius-Schur indicator: | $-1$ |
| Root number: | $-1$ |
| Artin field: | Galois closure of 8.8.359729184374784.2 |
| 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 \(\Q(\sqrt{6}, \sqrt{14})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 84x^{6} + 1890x^{4} - 10584x^{2} + 1764 \)
|
The roots of $f$ are computed in $\Q_{ 67 }$ to precision 10.
Roots:
| $r_{ 1 }$ | $=$ |
\( 13 + 62\cdot 67 + 27\cdot 67^{2} + 51\cdot 67^{3} + 5\cdot 67^{4} + 66\cdot 67^{5} + 36\cdot 67^{6} + 40\cdot 67^{7} + 3\cdot 67^{8} + 45\cdot 67^{9} +O(67^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 16 + 9\cdot 67 + 27\cdot 67^{2} + 2\cdot 67^{3} + 44\cdot 67^{4} + 14\cdot 67^{5} + 46\cdot 67^{6} + 25\cdot 67^{7} + 40\cdot 67^{8} + 63\cdot 67^{9} +O(67^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 17 + 48\cdot 67 + 66\cdot 67^{2} + 53\cdot 67^{3} + 55\cdot 67^{4} + 16\cdot 67^{5} + 49\cdot 67^{6} + 40\cdot 67^{7} + 15\cdot 67^{8} + 26\cdot 67^{9} +O(67^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 24 + 7\cdot 67 + 65\cdot 67^{2} + 14\cdot 67^{3} + 63\cdot 67^{4} + 32\cdot 67^{5} + 44\cdot 67^{6} + 56\cdot 67^{7} + 54\cdot 67^{8} + 21\cdot 67^{9} +O(67^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 43 + 59\cdot 67 + 67^{2} + 52\cdot 67^{3} + 3\cdot 67^{4} + 34\cdot 67^{5} + 22\cdot 67^{6} + 10\cdot 67^{7} + 12\cdot 67^{8} + 45\cdot 67^{9} +O(67^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 50 + 18\cdot 67 + 13\cdot 67^{3} + 11\cdot 67^{4} + 50\cdot 67^{5} + 17\cdot 67^{6} + 26\cdot 67^{7} + 51\cdot 67^{8} + 40\cdot 67^{9} +O(67^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 51 + 57\cdot 67 + 39\cdot 67^{2} + 64\cdot 67^{3} + 22\cdot 67^{4} + 52\cdot 67^{5} + 20\cdot 67^{6} + 41\cdot 67^{7} + 26\cdot 67^{8} + 3\cdot 67^{9} +O(67^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 54 + 4\cdot 67 + 39\cdot 67^{2} + 15\cdot 67^{3} + 61\cdot 67^{4} + 30\cdot 67^{6} + 26\cdot 67^{7} + 63\cdot 67^{8} + 21\cdot 67^{9} +O(67^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ | ✓ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | |
| $2$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $0$ | |
| $2$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $0$ | |
| $2$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ |