Basic invariants
| Dimension: | $1$ |
| Group: | $C_8$ |
| Conductor: | \(68\)\(\medspace = 2^{2} \cdot 17 \) |
| Artin field: | Galois closure of 8.0.105046700288.1 |
| Galois orbit size: | $4$ |
| Smallest permutation container: | $C_8$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{68}(15,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 17x^{6} + 68x^{4} + 85x^{2} + 17 \)
|
The roots of $f$ are computed in $\Q_{ 47 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 1 + 16\cdot 47 + 42\cdot 47^{2} + 9\cdot 47^{3} + 41\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 7 + 8\cdot 47 + 44\cdot 47^{2} + 39\cdot 47^{3} + 36\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 10 + 24\cdot 47 + 18\cdot 47^{2} + 23\cdot 47^{3} + 16\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 16 + 14\cdot 47 + 4\cdot 47^{2} + 35\cdot 47^{3} + 23\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 31 + 32\cdot 47 + 42\cdot 47^{2} + 11\cdot 47^{3} + 23\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 37 + 22\cdot 47 + 28\cdot 47^{2} + 23\cdot 47^{3} + 30\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 40 + 38\cdot 47 + 2\cdot 47^{2} + 7\cdot 47^{3} + 10\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 46 + 30\cdot 47 + 4\cdot 47^{2} + 37\cdot 47^{3} + 5\cdot 47^{4} +O(47^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-1$ |
| $1$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $\zeta_{8}^{2}$ |
| $1$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $-\zeta_{8}^{2}$ |
| $1$ | $8$ | $(1,5,2,6,8,4,7,3)$ | $-\zeta_{8}$ |
| $1$ | $8$ | $(1,6,7,5,8,3,2,4)$ | $-\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,4,2,3,8,5,7,6)$ | $\zeta_{8}$ |
| $1$ | $8$ | $(1,3,7,4,8,6,2,5)$ | $\zeta_{8}^{3}$ |
The blue line marks the conjugacy class containing complex conjugation.