Basic invariants
Dimension: | $1$ |
Group: | $C_8$ |
Conductor: | \(41\) |
Artin field: | Galois closure of 8.0.194754273881.1 |
Galois orbit size: | $4$ |
Smallest permutation container: | $C_8$ |
Parity: | odd |
Dirichlet character: | \(\chi_{41}(38,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - x^{7} + 3x^{6} - 11x^{5} + 44x^{4} + 53x^{3} + 153x^{2} + 160x + 59 \) . |
The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 7\cdot 59 + 43\cdot 59^{2} + 36\cdot 59^{3} + 7\cdot 59^{4} +O(59^{5})\)
$r_{ 2 }$ |
$=$ |
\( 18 + 7\cdot 59 + 53\cdot 59^{2} + 56\cdot 59^{3} + 57\cdot 59^{4} +O(59^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 32 + 15\cdot 59 + 17\cdot 59^{2} + 49\cdot 59^{3} + 48\cdot 59^{4} +O(59^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 37 + 52\cdot 59 + 17\cdot 59^{2} + 4\cdot 59^{4} +O(59^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 45 + 3\cdot 59 + 54\cdot 59^{2} + 8\cdot 59^{3} + 4\cdot 59^{4} +O(59^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 50 + 42\cdot 59 + 8\cdot 59^{2} + 27\cdot 59^{3} + 35\cdot 59^{4} +O(59^{5})\)
| $r_{ 7 }$ |
$=$ |
\( 56 + 43\cdot 59 + 41\cdot 59^{2} + 38\cdot 59^{3} + 35\cdot 59^{4} +O(59^{5})\)
| $r_{ 8 }$ |
$=$ |
\( 58 + 3\cdot 59 + 18\cdot 59^{3} + 42\cdot 59^{4} +O(59^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $1$ |
$1$ | $2$ | $(1,5)(2,7)(3,4)(6,8)$ | $-1$ |
$1$ | $4$ | $(1,6,5,8)(2,3,7,4)$ | $-\zeta_{8}^{2}$ |
$1$ | $4$ | $(1,8,5,6)(2,4,7,3)$ | $\zeta_{8}^{2}$ |
$1$ | $8$ | $(1,7,6,4,5,2,8,3)$ | $-\zeta_{8}^{3}$ |
$1$ | $8$ | $(1,4,8,7,5,3,6,2)$ | $-\zeta_{8}$ |
$1$ | $8$ | $(1,2,6,3,5,7,8,4)$ | $\zeta_{8}^{3}$ |
$1$ | $8$ | $(1,3,8,2,5,4,6,7)$ | $\zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.