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 8.0.1438916737499136.1 |
| 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 \(\Q(\sqrt{3}, \sqrt{14})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 84x^{6} + 1260x^{4} + 5292x^{2} + 441 \)
|
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 10.
Roots:
| $r_{ 1 }$ | $=$ |
\( 5 + 61 + 25\cdot 61^{2} + 16\cdot 61^{3} + 19\cdot 61^{4} + 55\cdot 61^{5} + 57\cdot 61^{6} + 61^{7} + 8\cdot 61^{8} + 38\cdot 61^{9} +O(61^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 7 + 23\cdot 61 + 3\cdot 61^{2} + 43\cdot 61^{3} + 2\cdot 61^{4} + 37\cdot 61^{5} + 23\cdot 61^{6} + 14\cdot 61^{7} + 39\cdot 61^{8} + 8\cdot 61^{9} +O(61^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 17 + 11\cdot 61 + 50\cdot 61^{2} + 11\cdot 61^{3} + 49\cdot 61^{4} + 27\cdot 61^{5} + 61^{6} + 29\cdot 61^{7} + 23\cdot 61^{8} + 48\cdot 61^{9} +O(61^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 23 + 40\cdot 61 + 61^{2} + 9\cdot 61^{3} + 48\cdot 61^{4} + 3\cdot 61^{5} + 35\cdot 61^{6} + 30\cdot 61^{7} + 41\cdot 61^{8} + 3\cdot 61^{9} +O(61^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 38 + 20\cdot 61 + 59\cdot 61^{2} + 51\cdot 61^{3} + 12\cdot 61^{4} + 57\cdot 61^{5} + 25\cdot 61^{6} + 30\cdot 61^{7} + 19\cdot 61^{8} + 57\cdot 61^{9} +O(61^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 44 + 49\cdot 61 + 10\cdot 61^{2} + 49\cdot 61^{3} + 11\cdot 61^{4} + 33\cdot 61^{5} + 59\cdot 61^{6} + 31\cdot 61^{7} + 37\cdot 61^{8} + 12\cdot 61^{9} +O(61^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 54 + 37\cdot 61 + 57\cdot 61^{2} + 17\cdot 61^{3} + 58\cdot 61^{4} + 23\cdot 61^{5} + 37\cdot 61^{6} + 46\cdot 61^{7} + 21\cdot 61^{8} + 52\cdot 61^{9} +O(61^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 56 + 59\cdot 61 + 35\cdot 61^{2} + 44\cdot 61^{3} + 41\cdot 61^{4} + 5\cdot 61^{5} + 3\cdot 61^{6} + 59\cdot 61^{7} + 52\cdot 61^{8} + 22\cdot 61^{9} +O(61^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | ✓ |
| $2$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $0$ | |
| $2$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $0$ | |
| $2$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ |