Basic invariants
| Dimension: | $1$ |
| Group: | $C_8$ |
| Conductor: | \(17\) |
| Artin field: | Galois closure of \(\Q(\zeta_{17})^+\) |
| Galois orbit size: | $4$ |
| Smallest permutation container: | $C_8$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{17}(9,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} - 7x^{6} + 6x^{5} + 15x^{4} - 10x^{3} - 10x^{2} + 4x + 1 \)
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The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 4 + 49\cdot 67 + 2\cdot 67^{2} + 56\cdot 67^{3} + 16\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 7 + 9\cdot 67 + 65\cdot 67^{2} + 22\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 20 + 7\cdot 67 + 12\cdot 67^{2} + 21\cdot 67^{3} + 5\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 34 + 55\cdot 67 + 10\cdot 67^{2} + 42\cdot 67^{3} + 58\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 45 + 47\cdot 67 + 37\cdot 67^{2} + 11\cdot 67^{3} + 50\cdot 67^{4} +O(67^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 52 + 61\cdot 67 + 57\cdot 67^{2} + 7\cdot 67^{3} + 53\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 53 + 9\cdot 67 + 56\cdot 67^{2} + 56\cdot 67^{3} + 64\cdot 67^{4} +O(67^{5})\)
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| $r_{ 8 }$ | $=$ |
\( 54 + 27\cdot 67 + 25\cdot 67^{2} + 4\cdot 67^{3} + 64\cdot 67^{4} +O(67^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,2)(3,7)(4,5)(6,8)$ | $-1$ |
| $1$ | $4$ | $(1,3,2,7)(4,8,5,6)$ | $\zeta_{8}^{2}$ |
| $1$ | $4$ | $(1,7,2,3)(4,6,5,8)$ | $-\zeta_{8}^{2}$ |
| $1$ | $8$ | $(1,8,3,5,2,6,7,4)$ | $-\zeta_{8}$ |
| $1$ | $8$ | $(1,5,7,8,2,4,3,6)$ | $-\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,6,3,4,2,8,7,5)$ | $\zeta_{8}$ |
| $1$ | $8$ | $(1,4,7,6,2,5,3,8)$ | $\zeta_{8}^{3}$ |
The blue line marks the conjugacy class containing complex conjugation.