Basic invariants
Dimension: | $1$ |
Group: | $C_8$ |
Conductor: | \(224\)\(\medspace = 2^{5} \cdot 7 \) |
Artin field: | Galois closure of 8.0.5156108238848.1 |
Galois orbit size: | $4$ |
Smallest permutation container: | $C_8$ |
Parity: | odd |
Dirichlet character: | \(\chi_{224}(125,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} + 56x^{6} + 980x^{4} + 5488x^{2} + 4802 \) . |
The roots of $f$ are computed in $\Q_{ 47 }$ to precision 7.
Roots:
$r_{ 1 }$ | $=$ | \( 1 + 23\cdot 47 + 24\cdot 47^{2} + 10\cdot 47^{3} + 45\cdot 47^{4} + 30\cdot 47^{5} + 5\cdot 47^{6} +O(47^{7})\) |
$r_{ 2 }$ | $=$ | \( 17 + 8\cdot 47 + 45\cdot 47^{2} + 23\cdot 47^{3} + 2\cdot 47^{4} + 9\cdot 47^{5} + 23\cdot 47^{6} +O(47^{7})\) |
$r_{ 3 }$ | $=$ | \( 21 + 30\cdot 47 + 19\cdot 47^{2} + 2\cdot 47^{3} + 12\cdot 47^{4} + 23\cdot 47^{5} + 17\cdot 47^{6} +O(47^{7})\) |
$r_{ 4 }$ | $=$ | \( 23 + 8\cdot 47 + 27\cdot 47^{2} + 22\cdot 47^{3} + 34\cdot 47^{4} + 20\cdot 47^{5} + 14\cdot 47^{6} +O(47^{7})\) |
$r_{ 5 }$ | $=$ | \( 24 + 38\cdot 47 + 19\cdot 47^{2} + 24\cdot 47^{3} + 12\cdot 47^{4} + 26\cdot 47^{5} + 32\cdot 47^{6} +O(47^{7})\) |
$r_{ 6 }$ | $=$ | \( 26 + 16\cdot 47 + 27\cdot 47^{2} + 44\cdot 47^{3} + 34\cdot 47^{4} + 23\cdot 47^{5} + 29\cdot 47^{6} +O(47^{7})\) |
$r_{ 7 }$ | $=$ | \( 30 + 38\cdot 47 + 47^{2} + 23\cdot 47^{3} + 44\cdot 47^{4} + 37\cdot 47^{5} + 23\cdot 47^{6} +O(47^{7})\) |
$r_{ 8 }$ | $=$ | \( 46 + 23\cdot 47 + 22\cdot 47^{2} + 36\cdot 47^{3} + 47^{4} + 16\cdot 47^{5} + 41\cdot 47^{6} +O(47^{7})\) |
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 |
$1$ | $1$ | $()$ | $1$ |
$1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-1$ |
$1$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $-\zeta_{8}^{2}$ |
$1$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $\zeta_{8}^{2}$ |
$1$ | $8$ | $(1,5,3,7,8,4,6,2)$ | $-\zeta_{8}^{3}$ |
$1$ | $8$ | $(1,7,6,5,8,2,3,4)$ | $-\zeta_{8}$ |
$1$ | $8$ | $(1,4,3,2,8,5,6,7)$ | $\zeta_{8}^{3}$ |
$1$ | $8$ | $(1,2,6,4,8,7,3,5)$ | $\zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.