Basic invariants
| Dimension: | $2$ |
| Group: | $C_4\wr C_2$ |
| Conductor: | \(624\)\(\medspace = 2^{4} \cdot 3 \cdot 13 \) |
| Artin stem field: | Galois closure of 8.0.45556992.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4\wr C_2$ |
| Parity: | odd |
| Determinant: | 1.156.4t1.a.b |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.2.105456.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 7x^{6} + 20x^{4} - 26x^{2} + 13 \)
|
The roots of $f$ are computed in $\Q_{ 43 }$ to precision 10.
Roots:
| $r_{ 1 }$ | $=$ |
\( 4 + 28\cdot 43 + 24\cdot 43^{2} + 14\cdot 43^{3} + 43^{4} + 4\cdot 43^{5} + 30\cdot 43^{6} + 24\cdot 43^{7} + 36\cdot 43^{8} + 14\cdot 43^{9} +O(43^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 6 + 21\cdot 43 + 5\cdot 43^{2} + 8\cdot 43^{3} + 21\cdot 43^{4} + 10\cdot 43^{5} + 13\cdot 43^{6} + 17\cdot 43^{7} + 4\cdot 43^{9} +O(43^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 14 + 6\cdot 43 + 12\cdot 43^{2} + 30\cdot 43^{3} + 25\cdot 43^{4} + 28\cdot 43^{5} + 30\cdot 43^{6} + 18\cdot 43^{7} + 29\cdot 43^{8} + 19\cdot 43^{9} +O(43^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 19 + 37\cdot 43 + 34\cdot 43^{2} + 33\cdot 43^{3} + 20\cdot 43^{4} + 22\cdot 43^{5} + 7\cdot 43^{6} + 43^{7} + 27\cdot 43^{8} + 39\cdot 43^{9} +O(43^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 24 + 5\cdot 43 + 8\cdot 43^{2} + 9\cdot 43^{3} + 22\cdot 43^{4} + 20\cdot 43^{5} + 35\cdot 43^{6} + 41\cdot 43^{7} + 15\cdot 43^{8} + 3\cdot 43^{9} +O(43^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 29 + 36\cdot 43 + 30\cdot 43^{2} + 12\cdot 43^{3} + 17\cdot 43^{4} + 14\cdot 43^{5} + 12\cdot 43^{6} + 24\cdot 43^{7} + 13\cdot 43^{8} + 23\cdot 43^{9} +O(43^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 37 + 21\cdot 43 + 37\cdot 43^{2} + 34\cdot 43^{3} + 21\cdot 43^{4} + 32\cdot 43^{5} + 29\cdot 43^{6} + 25\cdot 43^{7} + 42\cdot 43^{8} + 38\cdot 43^{9} +O(43^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 39 + 14\cdot 43 + 18\cdot 43^{2} + 28\cdot 43^{3} + 41\cdot 43^{4} + 38\cdot 43^{5} + 12\cdot 43^{6} + 18\cdot 43^{7} + 6\cdot 43^{8} + 28\cdot 43^{9} +O(43^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,8)(3,6)$ | $0$ |
| $4$ | $2$ | $(1,5)(2,3)(4,8)(6,7)$ | $0$ |
| $1$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $0$ |
| $2$ | $4$ | $(1,3,8,6)$ | $-\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,6,8,3)$ | $\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,8)(2,5,7,4)(3,6)$ | $\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,8)(2,4,7,5)(3,6)$ | $-\zeta_{4} - 1$ |
| $4$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ |
| $4$ | $8$ | $(1,7,6,4,8,2,3,5)$ | $0$ |
| $4$ | $8$ | $(1,4,3,7,8,5,6,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.