Basic invariants
| Dimension: | $1$ |
| Group: | $C_8$ |
| Conductor: | \(85\)\(\medspace = 5 \cdot 17 \) |
| Artin field: | Galois closure of 8.0.6411541765625.1 |
| Galois orbit size: | $4$ |
| Smallest permutation container: | $C_8$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{85}(8,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} + 10x^{6} - 79x^{5} + 134x^{4} + 41x^{3} + 245x^{2} - 846x + 596 \)
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The roots of $f$ are computed in $\Q_{ 83 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 7 + 39\cdot 83 + 3\cdot 83^{2} + 39\cdot 83^{3} + 72\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 15 + 71\cdot 83 + 51\cdot 83^{2} + 45\cdot 83^{3} + 21\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 19 + 56\cdot 83 + 63\cdot 83^{2} + 34\cdot 83^{3} + 53\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 42 + 24\cdot 83 + 21\cdot 83^{2} + 28\cdot 83^{3} + 81\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 44 + 37\cdot 83 + 37\cdot 83^{2} + 13\cdot 83^{3} + 36\cdot 83^{4} +O(83^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 52 + 67\cdot 83 + 45\cdot 83^{2} + 38\cdot 83^{3} + 14\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 75 + 50\cdot 83 + 73\cdot 83^{2} + 25\cdot 83^{4} +O(83^{5})\)
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| $r_{ 8 }$ | $=$ |
\( 79 + 67\cdot 83 + 34\cdot 83^{2} + 48\cdot 83^{3} + 27\cdot 83^{4} +O(83^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,3)(2,8)(4,7)(5,6)$ | $-1$ |
| $1$ | $4$ | $(1,8,3,2)(4,5,7,6)$ | $-\zeta_{8}^{2}$ |
| $1$ | $4$ | $(1,2,3,8)(4,6,7,5)$ | $\zeta_{8}^{2}$ |
| $1$ | $8$ | $(1,7,8,6,3,4,2,5)$ | $-\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,6,2,7,3,5,8,4)$ | $-\zeta_{8}$ |
| $1$ | $8$ | $(1,4,8,5,3,7,2,6)$ | $\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,5,2,4,3,6,8,7)$ | $\zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.