Basic invariants
| Dimension: | $1$ |
| Group: | $C_5$ |
| Conductor: | \(11\) |
| Artin number field: | Galois closure of \(\Q(\zeta_{11})^+\) |
| 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 $\Q_{ 23 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 9 + 20\cdot 23 + 11\cdot 23^{2} + 6\cdot 23^{3} + 11\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 12 + 11\cdot 23^{2} + 17\cdot 23^{3} + 20\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 13 + 4\cdot 23 + 7\cdot 23^{2} + 14\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 17 + 3\cdot 23 + 4\cdot 23^{2} + 4\cdot 23^{3} + 21\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 19 + 16\cdot 23 + 11\cdot 23^{2} + 17\cdot 23^{3} + 23^{4} +O(23^{5})\)
|
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 | |||
| $c1$ | $c2$ | $c3$ | $c4$ | |||
| $1$ | $1$ | $()$ | $1$ | $1$ | $1$ | $1$ |
| $1$ | $5$ | $(1,5,2,4,3)$ | $\zeta_{5}$ | $\zeta_{5}^{2}$ | $\zeta_{5}^{3}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
| $1$ | $5$ | $(1,2,3,5,4)$ | $\zeta_{5}^{2}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}$ | $\zeta_{5}^{3}$ |
| $1$ | $5$ | $(1,4,5,3,2)$ | $\zeta_{5}^{3}$ | $\zeta_{5}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}^{2}$ |
| $1$ | $5$ | $(1,3,4,2,5)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}^{3}$ | $\zeta_{5}^{2}$ | $\zeta_{5}$ |