Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8$ |
| Conductor: | \(142129\)\(\medspace = 13^{2} \cdot 29^{2} \) |
| Frobenius-Schur indicator: | $-1$ |
| Root number: | $-1$ |
| Artin field: | Galois closure of 8.0.2871098559212689.1 |
| 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 \(\Q(\sqrt{13}, \sqrt{29})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} + 72x^{6} - 592x^{5} + 3519x^{4} - 9968x^{3} + 23522x^{2} - 66393x + 114383 \)
|
The roots of $f$ are computed in $\Q_{ 103 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 5 + 42\cdot 103 + 20\cdot 103^{2} + 95\cdot 103^{3} + 60\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 14 + 64\cdot 103 + 79\cdot 103^{2} + 33\cdot 103^{3} + 40\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 15 + 2\cdot 103 + 101\cdot 103^{2} + 77\cdot 103^{3} + 7\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 17 + 87\cdot 103 + 96\cdot 103^{2} + 3\cdot 103^{3} + 44\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 25 + 82\cdot 103 + 36\cdot 103^{2} + 16\cdot 103^{3} + 36\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 58 + 66\cdot 103 + 10\cdot 103^{2} + 103^{3} + 78\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 88 + 31\cdot 103 + 14\cdot 103^{2} + 40\cdot 103^{3} + 31\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 90 + 35\cdot 103 + 52\cdot 103^{2} + 40\cdot 103^{3} + 10\cdot 103^{4} +O(103^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,4)(2,6)(3,8)(5,7)$ | $-2$ | ✓ |
| $2$ | $4$ | $(1,8,4,3)(2,5,6,7)$ | $0$ | |
| $2$ | $4$ | $(1,6,4,2)(3,7,8,5)$ | $0$ | |
| $2$ | $4$ | $(1,5,4,7)(2,3,6,8)$ | $0$ |