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 field: | Galois closure of 8.8.12696463968316569.1 |
| 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 \(\Q(\sqrt{21}, \sqrt{69})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} - 136x^{6} - 94x^{5} + 5029x^{4} + 6616x^{3} - 37504x^{2} + 14104x + 1360 \)
|
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 value | Complex conjugation |
| $1$ | $1$ | $()$ | $2$ | ✓ |
| $1$ | $2$ | $(1,4)(2,5)(3,8)(6,7)$ | $-2$ | |
| $2$ | $4$ | $(1,7,4,6)(2,3,5,8)$ | $0$ | |
| $2$ | $4$ | $(1,3,4,8)(2,6,5,7)$ | $0$ | |
| $2$ | $4$ | $(1,2,4,5)(3,7,8,6)$ | $0$ |