Basic invariants
| Dimension: | $1$ |
| Group: | $C_8$ |
| Conductor: | \(73\) |
| Artin field: | Galois closure of 8.0.11047398519097.1 |
| Galois orbit size: | $4$ |
| Smallest permutation container: | $C_8$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{73}(63,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} + 5x^{6} + 17x^{5} - 46x^{4} + 136x^{3} + 320x^{2} - 512x + 4096 \)
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The roots of $f$ are computed in $\Q_{ 37 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 14 + 35\cdot 37 + 20\cdot 37^{2} + 6\cdot 37^{3} + 24\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 24 + 9\cdot 37 + 2\cdot 37^{2} + 9\cdot 37^{3} + 12\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 25 + 10\cdot 37 + 26\cdot 37^{2} + 10\cdot 37^{3} + 14\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 27 + 19\cdot 37 + 12\cdot 37^{2} + 29\cdot 37^{3} + 25\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 29 + 5\cdot 37 + 11\cdot 37^{2} + 34\cdot 37^{3} + 30\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 33 + 5\cdot 37 + 19\cdot 37^{2} + 16\cdot 37^{3} + 27\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 35 + 33\cdot 37 + 22\cdot 37^{2} + 2\cdot 37^{3} + 13\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 36 + 26\cdot 37 + 32\cdot 37^{2} + 37^{3} +O(37^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $-1$ |
| $1$ | $4$ | $(1,3,4,2)(5,7,8,6)$ | $-\zeta_{8}^{2}$ |
| $1$ | $4$ | $(1,2,4,3)(5,6,8,7)$ | $\zeta_{8}^{2}$ |
| $1$ | $8$ | $(1,6,3,5,4,7,2,8)$ | $-\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,5,2,6,4,8,3,7)$ | $-\zeta_{8}$ |
| $1$ | $8$ | $(1,7,3,8,4,6,2,5)$ | $\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,8,2,7,4,5,3,6)$ | $\zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.