Basic invariants
| Dimension: | $2$ |
| Group: | $C_4\wr C_2$ |
| Conductor: | \(273\)\(\medspace = 3 \cdot 7 \cdot 13 \) |
| Artin stem field: | Galois closure of 8.0.8719893.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4\wr C_2$ |
| Parity: | odd |
| Determinant: | 1.273.4t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.2.322959.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} + 5x^{5} + x^{4} - 3x^{3} - 5x^{2} - 2x + 7 \)
|
The roots of $f$ are computed in $\Q_{ 103 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 7 + 97\cdot 103 + 86\cdot 103^{2} + 66\cdot 103^{3} + 43\cdot 103^{4} +O(103^{5})\)
| $r_{ 2 }$ |
$=$ |
\( 10 + 51\cdot 103 + 32\cdot 103^{2} + 35\cdot 103^{3} + 79\cdot 103^{4} +O(103^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 18 + 34\cdot 103 + 62\cdot 103^{2} + 66\cdot 103^{3} + 55\cdot 103^{4} +O(103^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 19 + 51\cdot 103 + 83\cdot 103^{2} + 89\cdot 103^{3} + 43\cdot 103^{4} +O(103^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 41 + 98\cdot 103 + 68\cdot 103^{2} + 28\cdot 103^{3} + 100\cdot 103^{4} +O(103^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 44 + 2\cdot 103 + 14\cdot 103^{2} + 22\cdot 103^{3} + 6\cdot 103^{4} +O(103^{5})\)
| $r_{ 7 }$ |
$=$ |
\( 81 + 38\cdot 103 + 102\cdot 103^{2} + 80\cdot 103^{3} + 17\cdot 103^{4} +O(103^{5})\)
| $r_{ 8 }$ |
$=$ |
\( 92 + 38\cdot 103 + 64\cdot 103^{2} + 21\cdot 103^{3} + 65\cdot 103^{4} +O(103^{5})\)
| |
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,7)(4,5)(6,8)$ | $-2$ |
| $2$ | $2$ | $(2,7)(4,5)$ | $0$ |
| $4$ | $2$ | $(1,2)(3,7)(4,6)(5,8)$ | $0$ |
| $1$ | $4$ | $(1,6,3,8)(2,4,7,5)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,8,3,6)(2,5,7,4)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(2,5,7,4)$ | $\zeta_{4} + 1$ |
| $2$ | $4$ | $(2,4,7,5)$ | $-\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,3)(2,4,7,5)(6,8)$ | $-\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,3)(2,5,7,4)(6,8)$ | $\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,6,3,8)(2,5,7,4)$ | $0$ |
| $4$ | $4$ | $(1,7,3,2)(4,6,5,8)$ | $0$ |
| $4$ | $8$ | $(1,5,8,7,3,4,6,2)$ | $0$ |
| $4$ | $8$ | $(1,7,6,5,3,2,8,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.