Basic invariants
Dimension: | $1$ |
Group: | $C_5$ |
Conductor: | \(31\) |
Artin number field: | Galois closure of 5.5.923521.1 |
Galois orbit size: | $4$ |
Smallest permutation container: | $C_5$ |
Parity: | even |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 37 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 10 + 20\cdot 37 + 28\cdot 37^{2} + 15\cdot 37^{3} + 5\cdot 37^{4} +O(37^{5})\) |
$r_{ 2 }$ | $=$ | \( 18 + 26\cdot 37 + 8\cdot 37^{2} + 19\cdot 37^{3} + 20\cdot 37^{4} +O(37^{5})\) |
$r_{ 3 }$ | $=$ | \( 20 + 8\cdot 37 + 17\cdot 37^{2} + 24\cdot 37^{3} + 4\cdot 37^{4} +O(37^{5})\) |
$r_{ 4 }$ | $=$ | \( 31 + 20\cdot 37 + 10\cdot 37^{2} + 27\cdot 37^{3} + 10\cdot 37^{4} +O(37^{5})\) |
$r_{ 5 }$ | $=$ | \( 33 + 34\cdot 37 + 8\cdot 37^{2} + 24\cdot 37^{3} + 32\cdot 37^{4} +O(37^{5})\) |
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 values | |||
$c1$ | $c2$ | $c3$ | $c4$ | |||
$1$ | $1$ | $()$ | $1$ | $1$ | $1$ | $1$ |
$1$ | $5$ | $(1,3,5,4,2)$ | $\zeta_{5}$ | $\zeta_{5}^{2}$ | $\zeta_{5}^{3}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
$1$ | $5$ | $(1,5,2,3,4)$ | $\zeta_{5}^{2}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}$ | $\zeta_{5}^{3}$ |
$1$ | $5$ | $(1,4,3,2,5)$ | $\zeta_{5}^{3}$ | $\zeta_{5}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}^{2}$ |
$1$ | $5$ | $(1,2,4,5,3)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}^{3}$ | $\zeta_{5}^{2}$ | $\zeta_{5}$ |