Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8$ |
| Conductor: | \(233289\)\(\medspace = 3^{2} \cdot 7^{2} \cdot 23^{2} \) |
| Frobenius-Schur indicator: | $-1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 8.8.12696463968316569.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $Q_8$ |
| Parity: | even |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{21}, \sqrt{69})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 17 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 2\cdot 17 + 4\cdot 17^{2} + 4\cdot 17^{3} + 9\cdot 17^{4} +O(17^{5})\)
| $r_{ 2 }$ |
$=$ |
\( 1 + 12\cdot 17 + 5\cdot 17^{2} + 3\cdot 17^{3} + 12\cdot 17^{4} +O(17^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 2 + 13\cdot 17 + 2\cdot 17^{2} + 4\cdot 17^{3} + 2\cdot 17^{4} +O(17^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 3 + 5\cdot 17 + 4\cdot 17^{2} + 14\cdot 17^{3} + 4\cdot 17^{4} +O(17^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 4 + 8\cdot 17 + 15\cdot 17^{2} + 13\cdot 17^{3} + 12\cdot 17^{4} +O(17^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 5 + 14\cdot 17 + 16\cdot 17^{2} + 5\cdot 17^{3} +O(17^{5})\)
| $r_{ 7 }$ |
$=$ |
\( 7 + 15\cdot 17 + 13\cdot 17^{2} + 8\cdot 17^{3} + 13\cdot 17^{4} +O(17^{5})\)
| $r_{ 8 }$ |
$=$ |
\( 13 + 14\cdot 17 + 4\cdot 17^{2} + 13\cdot 17^{3} + 12\cdot 17^{4} +O(17^{5})\)
| |
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 values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,5)(3,8)(6,7)$ | $-2$ |
| $2$ | $4$ | $(1,7,4,6)(2,3,5,8)$ | $0$ |
| $2$ | $4$ | $(1,3,4,8)(2,6,5,7)$ | $0$ |
| $2$ | $4$ | $(1,2,4,5)(3,7,8,6)$ | $0$ |