Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8$ |
| Conductor: | \(53361\)\(\medspace = 3^{2} \cdot 7^{2} \cdot 11^{2} \) |
| Frobenius-Schur indicator: | $-1$ |
| Root number: | $-1$ |
| Artin field: | Galois closure of 8.8.151939915084881.1 |
| 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 \(\Q(\sqrt{21}, \sqrt{33})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} - 62x^{6} + 66x^{5} + 1125x^{4} + 264x^{3} - 4982x^{2} - 4245x + 823 \)
|
The roots of $f$ are computed in $\Q_{ 167 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 6 + 140\cdot 167 + 18\cdot 167^{2} + 39\cdot 167^{3} + 151\cdot 167^{4} +O(167^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 9 + 11\cdot 167 + 160\cdot 167^{2} + 89\cdot 167^{3} + 105\cdot 167^{4} +O(167^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 22 + 119\cdot 167 + 72\cdot 167^{2} + 159\cdot 167^{3} + 117\cdot 167^{4} +O(167^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 29 + 159\cdot 167 + 119\cdot 167^{2} + 95\cdot 167^{3} + 154\cdot 167^{4} +O(167^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 31 + 4\cdot 167 + 82\cdot 167^{2} + 40\cdot 167^{3} + 164\cdot 167^{4} +O(167^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 88 + 14\cdot 167 + 73\cdot 167^{2} + 95\cdot 167^{3} + 119\cdot 167^{4} +O(167^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 154 + 152\cdot 167 + 91\cdot 167^{2} + 12\cdot 167^{3} + 84\cdot 167^{4} +O(167^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 165 + 66\cdot 167 + 49\cdot 167^{2} + 135\cdot 167^{3} + 104\cdot 167^{4} +O(167^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,3)(4,8)(5,6)$ | $-2$ |
| $2$ | $4$ | $(1,3,7,2)(4,6,8,5)$ | $0$ |
| $2$ | $4$ | $(1,8,7,4)(2,5,3,6)$ | $0$ |
| $2$ | $4$ | $(1,6,7,5)(2,8,3,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.