Basic invariants
| Dimension: | $1$ |
| Group: | $C_8$ |
| Conductor: | \(85\)\(\medspace = 5 \cdot 17 \) |
| Artin number field: | Galois closure of 8.0.6411541765625.2 |
| Galois orbit size: | $4$ |
| Smallest permutation container: | $C_8$ |
| Parity: | odd |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 4 + 13\cdot 43^{2} + 42\cdot 43^{3} + 39\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 13 + 14\cdot 43 + 22\cdot 43^{2} + 22\cdot 43^{3} + 13\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 23 + 29\cdot 43 + 41\cdot 43^{2} + 41\cdot 43^{3} + 43^{4} +O(43^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 24 + 32\cdot 43 + 21\cdot 43^{2} + 43^{3} + 29\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 36 + 27\cdot 43 + 21\cdot 43^{2} + 11\cdot 43^{3} + 16\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 37 + 31\cdot 43 + 3\cdot 43^{2} + 33\cdot 43^{3} + 3\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 39 + 40\cdot 43 + 35\cdot 43^{2} + 43^{3} + 8\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 40 + 37\cdot 43 + 11\cdot 43^{2} + 17\cdot 43^{3} + 16\cdot 43^{4} +O(43^{5})\)
|
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 values | |||
| $c1$ | $c2$ | $c3$ | $c4$ | |||
| $1$ | $1$ | $()$ | $1$ | $1$ | $1$ | $1$ |
| $1$ | $2$ | $(1,7)(2,5)(3,4)(6,8)$ | $-1$ | $-1$ | $-1$ | $-1$ |
| $1$ | $4$ | $(1,8,7,6)(2,4,5,3)$ | $\zeta_{8}^{2}$ | $-\zeta_{8}^{2}$ | $\zeta_{8}^{2}$ | $-\zeta_{8}^{2}$ |
| $1$ | $4$ | $(1,6,7,8)(2,3,5,4)$ | $-\zeta_{8}^{2}$ | $\zeta_{8}^{2}$ | $-\zeta_{8}^{2}$ | $\zeta_{8}^{2}$ |
| $1$ | $8$ | $(1,5,8,3,7,2,6,4)$ | $\zeta_{8}$ | $\zeta_{8}^{3}$ | $-\zeta_{8}$ | $-\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,3,6,5,7,4,8,2)$ | $\zeta_{8}^{3}$ | $\zeta_{8}$ | $-\zeta_{8}^{3}$ | $-\zeta_{8}$ |
| $1$ | $8$ | $(1,2,8,4,7,5,6,3)$ | $-\zeta_{8}$ | $-\zeta_{8}^{3}$ | $\zeta_{8}$ | $\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,4,6,2,7,3,8,5)$ | $-\zeta_{8}^{3}$ | $-\zeta_{8}$ | $\zeta_{8}^{3}$ | $\zeta_{8}$ |