Basic invariants
| Dimension: | $2$ |
| Group: | $C_4\wr C_2$ |
| Conductor: | \(2880\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \) |
| Artin stem field: | Galois closure of 8.0.23887872000.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4\wr C_2$ |
| Parity: | odd |
| Determinant: | 1.5.4t1.a.b |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.2.18000.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 6x^{4} + 45 \)
|
The roots of $f$ are computed in $\Q_{ 41 }$ to precision 10.
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 3\cdot 41 + 6\cdot 41^{2} + 33\cdot 41^{3} + 33\cdot 41^{4} + 20\cdot 41^{5} + 24\cdot 41^{6} + 34\cdot 41^{7} + 15\cdot 41^{8} + 26\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 12 + 33\cdot 41 + 33\cdot 41^{2} + 29\cdot 41^{3} + 19\cdot 41^{4} + 17\cdot 41^{5} + 27\cdot 41^{6} + 37\cdot 41^{7} + 10\cdot 41^{8} + 19\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 15 + 2\cdot 41 + 13\cdot 41^{2} + 40\cdot 41^{3} + 7\cdot 41^{4} + 25\cdot 41^{5} + 24\cdot 41^{6} + 8\cdot 41^{7} + 21\cdot 41^{8} + 14\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 18 + 4\cdot 41 + 27\cdot 41^{2} + 8\cdot 41^{3} + 22\cdot 41^{4} + 20\cdot 41^{5} + 34\cdot 41^{6} + 8\cdot 41^{7} + 33\cdot 41^{8} + 3\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 23 + 36\cdot 41 + 13\cdot 41^{2} + 32\cdot 41^{3} + 18\cdot 41^{4} + 20\cdot 41^{5} + 6\cdot 41^{6} + 32\cdot 41^{7} + 7\cdot 41^{8} + 37\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 26 + 38\cdot 41 + 27\cdot 41^{2} + 33\cdot 41^{4} + 15\cdot 41^{5} + 16\cdot 41^{6} + 32\cdot 41^{7} + 19\cdot 41^{8} + 26\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 29 + 7\cdot 41 + 7\cdot 41^{2} + 11\cdot 41^{3} + 21\cdot 41^{4} + 23\cdot 41^{5} + 13\cdot 41^{6} + 3\cdot 41^{7} + 30\cdot 41^{8} + 21\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 39 + 37\cdot 41 + 34\cdot 41^{2} + 7\cdot 41^{3} + 7\cdot 41^{4} + 20\cdot 41^{5} + 16\cdot 41^{6} + 6\cdot 41^{7} + 25\cdot 41^{8} + 14\cdot 41^{9} +O(41^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,8)(4,5)$ | $0$ |
| $4$ | $2$ | $(1,6)(2,4)(3,8)(5,7)$ | $0$ |
| $1$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ |
| $2$ | $4$ | $(1,5,8,4)$ | $\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,4,8,5)$ | $-\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,8)(2,3,7,6)(4,5)$ | $-\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,8)(2,6,7,3)(4,5)$ | $\zeta_{4} - 1$ |
| $4$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ |
| $4$ | $8$ | $(1,6,5,7,8,3,4,2)$ | $0$ |
| $4$ | $8$ | $(1,7,4,6,8,2,5,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.