Basic invariants
| Dimension: | $4$ |
| Group: | $Z_8 : Z_8^\times$ |
| Conductor: | \(2560000\)\(\medspace = 2^{12} \cdot 5^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.1024000000.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $Z_8 : Z_8^\times$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $C_2\times D_4$ |
| Projective stem field: | Galois closure of 8.0.6553600.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 5x^{4} - 25 \)
|
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 9.
Roots:
| $r_{ 1 }$ | $=$ |
\( 6 + 80\cdot 89 + 78\cdot 89^{2} + 4\cdot 89^{3} + 78\cdot 89^{4} + 40\cdot 89^{5} + 37\cdot 89^{6} + 20\cdot 89^{7} + 36\cdot 89^{8} +O(89^{9})\)
|
| $r_{ 2 }$ | $=$ |
\( 26 + 43\cdot 89 + 64\cdot 89^{2} + 63\cdot 89^{3} + 81\cdot 89^{4} + 15\cdot 89^{5} + 46\cdot 89^{6} + 86\cdot 89^{7} + 46\cdot 89^{8} +O(89^{9})\)
|
| $r_{ 3 }$ | $=$ |
\( 38 + 57\cdot 89 + 47\cdot 89^{2} + 27\cdot 89^{3} + 69\cdot 89^{4} + 23\cdot 89^{5} + 47\cdot 89^{6} + 19\cdot 89^{7} + 74\cdot 89^{8} +O(89^{9})\)
|
| $r_{ 4 }$ | $=$ |
\( 43 + 62\cdot 89 + 61\cdot 89^{2} + 8\cdot 89^{3} + 38\cdot 89^{4} + 24\cdot 89^{5} + 43\cdot 89^{6} + 41\cdot 89^{7} + 72\cdot 89^{8} +O(89^{9})\)
|
| $r_{ 5 }$ | $=$ |
\( 46 + 26\cdot 89 + 27\cdot 89^{2} + 80\cdot 89^{3} + 50\cdot 89^{4} + 64\cdot 89^{5} + 45\cdot 89^{6} + 47\cdot 89^{7} + 16\cdot 89^{8} +O(89^{9})\)
|
| $r_{ 6 }$ | $=$ |
\( 51 + 31\cdot 89 + 41\cdot 89^{2} + 61\cdot 89^{3} + 19\cdot 89^{4} + 65\cdot 89^{5} + 41\cdot 89^{6} + 69\cdot 89^{7} + 14\cdot 89^{8} +O(89^{9})\)
|
| $r_{ 7 }$ | $=$ |
\( 63 + 45\cdot 89 + 24\cdot 89^{2} + 25\cdot 89^{3} + 7\cdot 89^{4} + 73\cdot 89^{5} + 42\cdot 89^{6} + 2\cdot 89^{7} + 42\cdot 89^{8} +O(89^{9})\)
|
| $r_{ 8 }$ | $=$ |
\( 83 + 8\cdot 89 + 10\cdot 89^{2} + 84\cdot 89^{3} + 10\cdot 89^{4} + 48\cdot 89^{5} + 51\cdot 89^{6} + 68\cdot 89^{7} + 52\cdot 89^{8} +O(89^{9})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-4$ |
| $2$ | $2$ | $(1,8)(2,7)$ | $0$ |
| $4$ | $2$ | $(1,7)(2,8)(4,5)$ | $0$ |
| $4$ | $2$ | $(1,8)(3,4)(5,6)$ | $0$ |
| $4$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $0$ |
| $2$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $0$ |
| $4$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $0$ |
| $4$ | $8$ | $(1,6,2,5,8,3,7,4)$ | $0$ |
| $4$ | $8$ | $(1,6,7,4,8,3,2,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.