Basic invariants
Dimension: | $1$ |
Group: | $C_5$ |
Conductor: | \(101\) |
Artin field: | Galois closure of 5.5.104060401.1 |
Galois orbit size: | $4$ |
Smallest permutation container: | $C_5$ |
Parity: | even |
Dirichlet character: | \(\chi_{101}(95,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - x^{4} - 40x^{3} - 93x^{2} - 21x + 17 \) . |
The roots of $f$ are computed in $\Q_{ 41 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 6 + 30\cdot 41 + 13\cdot 41^{2} + 2\cdot 41^{3} + 8\cdot 41^{4} +O(41^{5})\) |
$r_{ 2 }$ | $=$ | \( 21 + 36\cdot 41 + 33\cdot 41^{2} + 22\cdot 41^{3} + 19\cdot 41^{4} +O(41^{5})\) |
$r_{ 3 }$ | $=$ | \( 24 + 35\cdot 41 + 15\cdot 41^{3} + 39\cdot 41^{4} +O(41^{5})\) |
$r_{ 4 }$ | $=$ | \( 34 + 8\cdot 41 + 26\cdot 41^{2} + 29\cdot 41^{3} + 10\cdot 41^{4} +O(41^{5})\) |
$r_{ 5 }$ | $=$ | \( 39 + 11\cdot 41 + 7\cdot 41^{2} + 12\cdot 41^{3} + 4\cdot 41^{4} +O(41^{5})\) |
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 |
$1$ | $1$ | $()$ | $1$ |
$1$ | $5$ | $(1,5,2,4,3)$ | $\zeta_{5}^{2}$ |
$1$ | $5$ | $(1,2,3,5,4)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
$1$ | $5$ | $(1,4,5,3,2)$ | $\zeta_{5}$ |
$1$ | $5$ | $(1,3,4,2,5)$ | $\zeta_{5}^{3}$ |
The blue line marks the conjugacy class containing complex conjugation.