Basic invariants
| Dimension: | $1$ |
| Group: | $C_8$ |
| Conductor: | \(51\)\(\medspace = 3 \cdot 17 \) |
| Artin field: | Galois closure of 8.0.33237432513.1 |
| Galois orbit size: | $4$ |
| Smallest permutation container: | $C_8$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{51}(32,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} + 10x^{6} - 11x^{5} + 15x^{4} - 61x^{3} + 58x^{2} - 47x + 103 \)
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The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 1 + 7\cdot 67 + 61\cdot 67^{2} + 65\cdot 67^{3} + 58\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 5 + 10\cdot 67 + 7\cdot 67^{2} + 12\cdot 67^{3} + 11\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 27 + 19\cdot 67 + 22\cdot 67^{3} + 21\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 29 + 64\cdot 67 + 42\cdot 67^{2} + 13\cdot 67^{3} + 24\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 45 + 60\cdot 67^{2} + 34\cdot 67^{3} + 17\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 50 + 38\cdot 67 + 5\cdot 67^{2} + 40\cdot 67^{3} + 17\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 51 + 38\cdot 67 + 35\cdot 67^{3} + 17\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 61 + 21\cdot 67 + 23\cdot 67^{2} + 44\cdot 67^{3} + 32\cdot 67^{4} +O(67^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,2)(3,7)(4,6)(5,8)$ | $-1$ |
| $1$ | $4$ | $(1,7,2,3)(4,5,6,8)$ | $\zeta_{8}^{2}$ |
| $1$ | $4$ | $(1,3,2,7)(4,8,6,5)$ | $-\zeta_{8}^{2}$ |
| $1$ | $8$ | $(1,5,7,6,2,8,3,4)$ | $\zeta_{8}$ |
| $1$ | $8$ | $(1,6,3,5,2,4,7,8)$ | $\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,8,7,4,2,5,3,6)$ | $-\zeta_{8}$ |
| $1$ | $8$ | $(1,4,3,8,2,6,7,5)$ | $-\zeta_{8}^{3}$ |
The blue line marks the conjugacy class containing complex conjugation.