Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8$ |
| Conductor: | \(48841\)\(\medspace = 13^{2} \cdot 17^{2} \) |
| Frobenius-Schur indicator: | $-1$ |
| Root number: | $-1$ |
| Artin field: | Galois closure of 8.8.116507435287321.1 |
| 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 \(\Q(\sqrt{13}, \sqrt{17})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} - 67x^{6} - 16x^{5} + 863x^{4} + 1276x^{3} + 392x^{2} - 54x + 1 \)
|
The roots of $f$ are computed in $\Q_{ 179 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 42 + 54\cdot 179 + 111\cdot 179^{2} + 6\cdot 179^{3} + 165\cdot 179^{4} +O(179^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 46 + 76\cdot 179 + 151\cdot 179^{2} + 6\cdot 179^{3} + 114\cdot 179^{4} +O(179^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 49 + 122\cdot 179 + 153\cdot 179^{2} + 19\cdot 179^{3} + 124\cdot 179^{4} +O(179^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 50 + 156\cdot 179 + 2\cdot 179^{2} + 169\cdot 179^{3} + 73\cdot 179^{4} +O(179^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 101 + 155\cdot 179 + 92\cdot 179^{2} + 101\cdot 179^{3} + 139\cdot 179^{4} +O(179^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 136 + 98\cdot 179 + 164\cdot 179^{2} + 124\cdot 179^{3} + 100\cdot 179^{4} +O(179^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 142 + 113\cdot 179 + 82\cdot 179^{2} + 47\cdot 179^{3} + 177\cdot 179^{4} +O(179^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 153 + 117\cdot 179 + 135\cdot 179^{2} + 60\cdot 179^{3} +O(179^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ | ✓ |
| $1$ | $2$ | $(1,4)(2,7)(3,6)(5,8)$ | $-2$ | |
| $2$ | $4$ | $(1,6,4,3)(2,5,7,8)$ | $0$ | |
| $2$ | $4$ | $(1,2,4,7)(3,5,6,8)$ | $0$ | |
| $2$ | $4$ | $(1,8,4,5)(2,6,7,3)$ | $0$ |