Basic invariants
Dimension: | $1$ |
Group: | $C_5$ |
Conductor: | \(571\) |
Artin field: | Galois closure of 5.5.106302733681.1 |
Galois orbit size: | $4$ |
Smallest permutation container: | $C_5$ |
Parity: | even |
Dirichlet character: | \(\chi_{571}(167,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - x^{4} - 228x^{3} - 868x^{2} + 3056x + 7552 \) . |
The roots of $f$ are computed in $\Q_{ 29 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 1 + 9\cdot 29 + 14\cdot 29^{2} + 16\cdot 29^{3} + 16\cdot 29^{4} +O(29^{5})\) |
$r_{ 2 }$ | $=$ | \( 17 + 3\cdot 29 + 18\cdot 29^{2} + 10\cdot 29^{3} + 20\cdot 29^{4} +O(29^{5})\) |
$r_{ 3 }$ | $=$ | \( 21 + 29 + 24\cdot 29^{2} + 9\cdot 29^{3} + 21\cdot 29^{4} +O(29^{5})\) |
$r_{ 4 }$ | $=$ | \( 23 + 8\cdot 29 + 4\cdot 29^{2} + 12\cdot 29^{4} +O(29^{5})\) |
$r_{ 5 }$ | $=$ | \( 26 + 5\cdot 29 + 26\cdot 29^{2} + 20\cdot 29^{3} + 16\cdot 29^{4} +O(29^{5})\) |
Generators of the action on the roots $r_1, \ldots, r_{ 5 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 5 }$ | Character value |
$1$ | $1$ | $()$ | $1$ |
$1$ | $5$ | $(1,3,5,2,4)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
$1$ | $5$ | $(1,5,4,3,2)$ | $\zeta_{5}^{3}$ |
$1$ | $5$ | $(1,2,3,4,5)$ | $\zeta_{5}^{2}$ |
$1$ | $5$ | $(1,4,2,5,3)$ | $\zeta_{5}$ |
The blue line marks the conjugacy class containing complex conjugation.