Basic invariants
| Dimension: | $4$ |
| Group: | $Z_8 : Z_8^\times$ |
| Conductor: | \(115600\)\(\medspace = 2^{4} \cdot 5^{2} \cdot 17^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.31443200.2 |
| 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 8.0.46240000.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{6} - 4x^{5} - 2x^{4} + 4x^{3} + 12x^{2} + 6x + 1 \)
|
The roots of $f$ are computed in $\Q_{ 409 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 81 + 183\cdot 409 + 53\cdot 409^{2} + 136\cdot 409^{3} + 203\cdot 409^{4} +O(409^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 91 + 238\cdot 409 + 53\cdot 409^{2} + 214\cdot 409^{3} + 373\cdot 409^{4} +O(409^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 181 + 184\cdot 409 + 129\cdot 409^{2} + 321\cdot 409^{3} + 380\cdot 409^{4} +O(409^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 274 + 314\cdot 409 + 111\cdot 409^{2} + 311\cdot 409^{3} + 103\cdot 409^{4} +O(409^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 311 + 309\cdot 409 + 275\cdot 409^{2} + 169\cdot 409^{3} + 48\cdot 409^{4} +O(409^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 323 + 44\cdot 409 + 193\cdot 409^{2} + 335\cdot 409^{3} + 361\cdot 409^{4} +O(409^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 388 + 34\cdot 409 + 74\cdot 409^{2} + 334\cdot 409^{3} + 403\cdot 409^{4} +O(409^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 396 + 325\cdot 409 + 335\cdot 409^{2} + 222\cdot 409^{3} + 169\cdot 409^{4} +O(409^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,5)(2,4)(3,7)(6,8)$ | $-4$ |
| $2$ | $2$ | $(2,4)(6,8)$ | $0$ |
| $4$ | $2$ | $(1,4)(2,5)(3,8)(6,7)$ | $0$ |
| $4$ | $2$ | $(2,6)(3,7)(4,8)$ | $0$ |
| $4$ | $2$ | $(1,7)(3,5)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,7,5,3)(2,8,4,6)$ | $0$ |
| $2$ | $4$ | $(1,7,5,3)(2,6,4,8)$ | $0$ |
| $4$ | $4$ | $(1,2,5,4)(3,6,7,8)$ | $0$ |
| $4$ | $8$ | $(1,8,7,2,5,6,3,4)$ | $0$ |
| $4$ | $8$ | $(1,8,3,4,5,6,7,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.