Basic invariants
| Dimension: | $1$ |
| Group: | $C_5$ |
| Conductor: | \(25\)\(\medspace = 5^{2} \) |
| Artin number field: | Galois closure of 5.5.390625.1 |
| 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 $\Q_{ 43 }$ 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 values | |||
| $c1$ | $c2$ | $c3$ | $c4$ | |||
| $1$ | $1$ | $()$ | $1$ | $1$ | $1$ | $1$ |
| $1$ | $5$ | $(1,2,4,3,5)$ | $\zeta_{5}$ | $\zeta_{5}^{2}$ | $\zeta_{5}^{3}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
| $1$ | $5$ | $(1,4,5,2,3)$ | $\zeta_{5}^{2}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}$ | $\zeta_{5}^{3}$ |
| $1$ | $5$ | $(1,3,2,5,4)$ | $\zeta_{5}^{3}$ | $\zeta_{5}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}^{2}$ |
| $1$ | $5$ | $(1,5,3,4,2)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}^{3}$ | $\zeta_{5}^{2}$ | $\zeta_{5}$ |