Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8$ |
| Conductor: | \(119025\)\(\medspace = 3^{2} \cdot 5^{2} \cdot 23^{2} \) |
| Frobenius-Schur indicator: | $-1$ |
| Root number: | $-1$ |
| Artin field: | Galois closure of 8.0.1686221298140625.1 |
| 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 \(\Q(\sqrt{5}, \sqrt{69})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} + 64x^{6} - 192x^{5} + 1836x^{4} - 4635x^{3} + 39520x^{2} - 110850x + 565795 \)
|
The roots of $f$ are computed in $\Q_{ 139 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 5 + 7\cdot 139 + 19\cdot 139^{2} + 56\cdot 139^{3} + 131\cdot 139^{4} +O(139^{5})\)
| $r_{ 2 }$ |
$=$ |
\( 19 + 73\cdot 139 + 25\cdot 139^{2} + 93\cdot 139^{3} + 78\cdot 139^{4} +O(139^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 30 + 53\cdot 139 + 93\cdot 139^{3} + 93\cdot 139^{4} +O(139^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 69 + 30\cdot 139 + 101\cdot 139^{2} + 94\cdot 139^{3} + 2\cdot 139^{4} +O(139^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 95 + 87\cdot 139 + 76\cdot 139^{2} + 116\cdot 139^{3} + 56\cdot 139^{4} +O(139^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 101 + 19\cdot 139 + 7\cdot 139^{2} + 85\cdot 139^{3} + 113\cdot 139^{4} +O(139^{5})\)
| $r_{ 7 }$ |
$=$ |
\( 117 + 131\cdot 139 + 83\cdot 139^{2} + 53\cdot 139^{3} + 10\cdot 139^{4} +O(139^{5})\)
| $r_{ 8 }$ |
$=$ |
\( 123 + 13\cdot 139 + 103\cdot 139^{2} + 102\cdot 139^{3} + 68\cdot 139^{4} +O(139^{5})\)
| |
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,8)(3,5)(6,7)$ | $-2$ |
| $2$ | $4$ | $(1,2,4,8)(3,7,5,6)$ | $0$ |
| $2$ | $4$ | $(1,6,4,7)(2,5,8,3)$ | $0$ |
| $2$ | $4$ | $(1,5,4,3)(2,7,8,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.