Basic invariants
| Dimension: | $1$ |
| Group: | $C_8$ |
| Conductor: | \(85\)\(\medspace = 5 \cdot 17 \) |
| Artin field: | Galois closure of 8.8.256461670625.1 |
| Galois orbit size: | $4$ |
| Smallest permutation container: | $C_8$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{85}(9,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} - 24x^{6} + 23x^{5} + 151x^{4} - 197x^{3} - 214x^{2} + 429x - 169 \)
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The roots of $f$ are computed in $\Q_{ 101 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 20 + 45\cdot 101 + 52\cdot 101^{2} + 19\cdot 101^{3} + 26\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 21 + 49\cdot 101 + 84\cdot 101^{2} + 10\cdot 101^{3} + 75\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 32 + 101 + 101^{2} + 48\cdot 101^{3} + 55\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 35 + 23\cdot 101 + 29\cdot 101^{2} + 55\cdot 101^{3} + 101^{4} +O(101^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 46 + 80\cdot 101 + 45\cdot 101^{2} + 83\cdot 101^{3} + 7\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 70 + 45\cdot 101 + 89\cdot 101^{2} + 30\cdot 101^{3} + 44\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 87 + 35\cdot 101 + 54\cdot 101^{2} + 81\cdot 101^{3} + 14\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 94 + 21\cdot 101 + 47\cdot 101^{2} + 74\cdot 101^{3} + 77\cdot 101^{4} +O(101^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,2)(3,8)(4,6)(5,7)$ | $-1$ |
| $1$ | $4$ | $(1,5,2,7)(3,4,8,6)$ | $\zeta_{8}^{2}$ |
| $1$ | $4$ | $(1,7,2,5)(3,6,8,4)$ | $-\zeta_{8}^{2}$ |
| $1$ | $8$ | $(1,4,5,8,2,6,7,3)$ | $-\zeta_{8}$ |
| $1$ | $8$ | $(1,8,7,4,2,3,5,6)$ | $-\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,6,5,3,2,4,7,8)$ | $\zeta_{8}$ |
| $1$ | $8$ | $(1,3,7,6,2,8,5,4)$ | $\zeta_{8}^{3}$ |
The blue line marks the conjugacy class containing complex conjugation.