Basic invariants
Dimension: | $1$ |
Group: | $C_5$ |
Conductor: | \(25\)\(\medspace = 5^{2} \) |
Artin field: | Galois closure of 5.5.390625.1 |
Galois orbit size: | $4$ |
Smallest permutation container: | $C_5$ |
Parity: | even |
Dirichlet character: | \(\chi_{25}(6,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - 10x^{3} - 5x^{2} + 10x - 1 \) . |
The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 3 + 3\cdot 43 + 24\cdot 43^{2} + 5\cdot 43^{3} + 7\cdot 43^{4} +O(43^{5})\) |
$r_{ 2 }$ | $=$ | \( 7 + 13\cdot 43 + 19\cdot 43^{2} + 5\cdot 43^{3} + 21\cdot 43^{4} +O(43^{5})\) |
$r_{ 3 }$ | $=$ | \( 19 + 14\cdot 43 + 26\cdot 43^{2} + 32\cdot 43^{3} + 5\cdot 43^{4} +O(43^{5})\) |
$r_{ 4 }$ | $=$ | \( 24 + 12\cdot 43 + 2\cdot 43^{2} + 38\cdot 43^{3} + 26\cdot 43^{4} +O(43^{5})\) |
$r_{ 5 }$ | $=$ | \( 33 + 42\cdot 43 + 13\cdot 43^{2} + 4\cdot 43^{3} + 25\cdot 43^{4} +O(43^{5})\) |
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 |
$1$ | $1$ | $()$ | $1$ |
$1$ | $5$ | $(1,2,4,3,5)$ | $\zeta_{5}^{3}$ |
$1$ | $5$ | $(1,4,5,2,3)$ | $\zeta_{5}$ |
$1$ | $5$ | $(1,3,2,5,4)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
$1$ | $5$ | $(1,5,3,4,2)$ | $\zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.