Basic invariants
| Dimension: | $2$ |
| Group: | $C_4\wr C_2$ |
| Conductor: | \(320\)\(\medspace = 2^{6} \cdot 5 \) |
| Artin stem field: | Galois closure of 8.0.32768000.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_4\wr C_2$ |
| Parity: | odd |
| Determinant: | 1.5.4t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.2.2000.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 2x^{4} + 5 \)
|
The roots of $f$ are computed in $\Q_{ 101 }$ to precision 7.
Roots:
| $r_{ 1 }$ | $=$ |
\( 5 + 80\cdot 101 + 6\cdot 101^{2} + 81\cdot 101^{3} + 82\cdot 101^{4} + 64\cdot 101^{5} + 26\cdot 101^{6} +O(101^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 27 + 93\cdot 101 + 60\cdot 101^{2} + 4\cdot 101^{3} + 88\cdot 101^{4} + 2\cdot 101^{5} + 33\cdot 101^{6} +O(101^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 33 + 43\cdot 101 + 60\cdot 101^{2} + 10\cdot 101^{3} + 95\cdot 101^{4} + 97\cdot 101^{5} + 80\cdot 101^{6} +O(101^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 50 + 17\cdot 101 + 7\cdot 101^{2} + 62\cdot 101^{3} + 97\cdot 101^{4} + 64\cdot 101^{5} + 22\cdot 101^{6} +O(101^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 51 + 83\cdot 101 + 93\cdot 101^{2} + 38\cdot 101^{3} + 3\cdot 101^{4} + 36\cdot 101^{5} + 78\cdot 101^{6} +O(101^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 68 + 57\cdot 101 + 40\cdot 101^{2} + 90\cdot 101^{3} + 5\cdot 101^{4} + 3\cdot 101^{5} + 20\cdot 101^{6} +O(101^{7})\)
|
| $r_{ 7 }$ | $=$ |
\( 74 + 7\cdot 101 + 40\cdot 101^{2} + 96\cdot 101^{3} + 12\cdot 101^{4} + 98\cdot 101^{5} + 67\cdot 101^{6} +O(101^{7})\)
|
| $r_{ 8 }$ | $=$ |
\( 96 + 20\cdot 101 + 94\cdot 101^{2} + 19\cdot 101^{3} + 18\cdot 101^{4} + 36\cdot 101^{5} + 74\cdot 101^{6} +O(101^{7})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(2,7)(3,6)$ | $0$ |
| $4$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $1$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(2,3,7,6)$ | $\zeta_{4} + 1$ |
| $2$ | $4$ | $(2,6,7,3)$ | $-\zeta_{4} + 1$ |
| $2$ | $4$ | $(1,8)(2,6,7,3)(4,5)$ | $-\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,8)(2,3,7,6)(4,5)$ | $\zeta_{4} - 1$ |
| $2$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ |
| $4$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $0$ |
| $4$ | $8$ | $(1,3,4,7,8,6,5,2)$ | $0$ |
| $4$ | $8$ | $(1,7,5,3,8,2,4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.