Basic invariants
| Dimension: | $1$ |
| Group: | $C_8$ |
| Conductor: | \(32\)\(\medspace = 2^{5} \) |
| Artin field: | Galois closure of 8.0.2147483648.1 |
| Galois orbit size: | $4$ |
| Smallest permutation container: | $C_8$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{32}(27,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 8x^{6} + 20x^{4} + 16x^{2} + 2 \)
|
The roots of $f$ are computed in $\Q_{ 47 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 1 + 40\cdot 47 + 5\cdot 47^{2} + 20\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 4 + 5\cdot 47 + 41\cdot 47^{2} + 19\cdot 47^{3} + 16\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 11 + 13\cdot 47 + 8\cdot 47^{2} + 22\cdot 47^{3} + 20\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 18 + 25\cdot 47 + 45\cdot 47^{3} + 35\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 29 + 21\cdot 47 + 46\cdot 47^{2} + 47^{3} + 11\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 36 + 33\cdot 47 + 38\cdot 47^{2} + 24\cdot 47^{3} + 26\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 43 + 41\cdot 47 + 5\cdot 47^{2} + 27\cdot 47^{3} + 30\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 46 + 6\cdot 47 + 41\cdot 47^{2} + 46\cdot 47^{3} + 26\cdot 47^{4} +O(47^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-1$ |
| $1$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $-\zeta_{8}^{2}$ |
| $1$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $\zeta_{8}^{2}$ |
| $1$ | $8$ | $(1,7,5,3,8,2,4,6)$ | $\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,3,4,7,8,6,5,2)$ | $\zeta_{8}$ |
| $1$ | $8$ | $(1,2,5,6,8,7,4,3)$ | $-\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,6,4,2,8,3,5,7)$ | $-\zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.