Basic invariants
| Dimension: | $1$ |
| Group: | $C_8$ |
| Conductor: | \(89\) |
| Artin field: | Galois closure of 8.0.44231334895529.1 |
| Galois orbit size: | $4$ |
| Smallest permutation container: | $C_8$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{89}(52,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} + 6x^{6} - 46x^{5} - 143x^{4} + 575x^{3} + 1160x^{2} - 16x + 512 \)
|
The roots of $f$ are computed in $\Q_{ 97 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 41\cdot 97 + 84\cdot 97^{2} + 84\cdot 97^{3} + 29\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 31 + 45\cdot 97 + 20\cdot 97^{2} + 76\cdot 97^{3} + 81\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 60 + 17\cdot 97 + 84\cdot 97^{2} + 62\cdot 97^{3} + 4\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 67 + 6\cdot 97 + 69\cdot 97^{2} + 12\cdot 97^{3} + 38\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 68 + 33\cdot 97 + 44\cdot 97^{2} + 11\cdot 97^{3} + 26\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 81 + 33\cdot 97 + 56\cdot 97^{2} + 19\cdot 97^{3} + 95\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 83 + 8\cdot 97 + 73\cdot 97^{2} + 7\cdot 97^{3} + 37\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 93 + 6\cdot 97 + 53\cdot 97^{2} + 15\cdot 97^{3} + 75\cdot 97^{4} +O(97^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,7)(2,5)(3,4)(6,8)$ | $-1$ |
| $1$ | $4$ | $(1,6,7,8)(2,4,5,3)$ | $-\zeta_{8}^{2}$ |
| $1$ | $4$ | $(1,8,7,6)(2,3,5,4)$ | $\zeta_{8}^{2}$ |
| $1$ | $8$ | $(1,5,6,3,7,2,8,4)$ | $-\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,3,8,5,7,4,6,2)$ | $-\zeta_{8}$ |
| $1$ | $8$ | $(1,2,6,4,7,5,8,3)$ | $\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,4,8,2,7,3,6,5)$ | $\zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.