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 8.0.12696463968316569.1 |
| 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
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} + 109x^{6} + 138x^{5} + 3801x^{4} + 13938x^{3} + 54538x^{2} + 67350x + 58153 \)
|
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 1 + 29\cdot 89^{2} + 7\cdot 89^{3} + 46\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 10 + 72\cdot 89 + 71\cdot 89^{2} + 47\cdot 89^{3} + 20\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 22 + 79\cdot 89 + 70\cdot 89^{2} + 39\cdot 89^{3} + 39\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 27 + 79\cdot 89 + 14\cdot 89^{2} + 2\cdot 89^{3} + 29\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 36 + 83\cdot 89 + 31\cdot 89^{2} + 62\cdot 89^{3} + 3\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 48 + 23\cdot 89 + 17\cdot 89^{2} + 50\cdot 89^{3} + 77\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 53 + 26\cdot 89 + 55\cdot 89^{2} + 7\cdot 89^{3} + 44\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 73 + 80\cdot 89 + 64\cdot 89^{2} + 49\cdot 89^{3} + 6\cdot 89^{4} +O(89^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,6)(2,7)(3,5)(4,8)$ | $-2$ | ✓ |
| $2$ | $4$ | $(1,4,6,8)(2,3,7,5)$ | $0$ | |
| $2$ | $4$ | $(1,7,6,2)(3,8,5,4)$ | $0$ | |
| $2$ | $4$ | $(1,3,6,5)(2,8,7,4)$ | $0$ |