Basic invariants
| Dimension: | $1$ |
| Group: | $C_5$ |
| Conductor: | \(31\) |
| Artin field: | Galois closure of 5.5.923521.1 |
| Galois orbit size: | $4$ |
| Smallest permutation container: | $C_5$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{31}(16,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{4} - 12x^{3} + 21x^{2} + x - 5 \)
|
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 value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $5$ | $(1,3,5,4,2)$ | $\zeta_{5}^{3}$ |
| $1$ | $5$ | $(1,5,2,3,4)$ | $\zeta_{5}$ |
| $1$ | $5$ | $(1,4,3,2,5)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
| $1$ | $5$ | $(1,2,4,5,3)$ | $\zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.