# Properties

 Label 1.11.5t1.a Dimension $1$ Group $C_5$ Conductor $11$ Indicator $0$

# Related objects

## 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:
 $r_{ 1 }$ $=$ $$9 + 20\cdot 23 + 11\cdot 23^{2} + 6\cdot 23^{3} + 11\cdot 23^{4} +O(23^{5})$$ 9 + 20*23 + 11*23^2 + 6*23^3 + 11*23^4+O(23^5) $r_{ 2 }$ $=$ $$12 + 11\cdot 23^{2} + 17\cdot 23^{3} + 20\cdot 23^{4} +O(23^{5})$$ 12 + 11*23^2 + 17*23^3 + 20*23^4+O(23^5) $r_{ 3 }$ $=$ $$13 + 4\cdot 23 + 7\cdot 23^{2} + 14\cdot 23^{4} +O(23^{5})$$ 13 + 4*23 + 7*23^2 + 14*23^4+O(23^5) $r_{ 4 }$ $=$ $$17 + 3\cdot 23 + 4\cdot 23^{2} + 4\cdot 23^{3} + 21\cdot 23^{4} +O(23^{5})$$ 17 + 3*23 + 4*23^2 + 4*23^3 + 21*23^4+O(23^5) $r_{ 5 }$ $=$ $$19 + 16\cdot 23 + 11\cdot 23^{2} + 17\cdot 23^{3} + 23^{4} +O(23^{5})$$ 19 + 16*23 + 11*23^2 + 17*23^3 + 23^4+O(23^5)

### Generators of the action on the roots $r_1, \ldots, r_{ 5 }$

 Cycle notation $(1,5,2,4,3)$

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 5 }$ Character values $c1$ $c2$ $c3$ $c4$ $1$ $1$ $()$ $1$ $1$ $1$ $1$ $1$ $5$ $(1,5,2,4,3)$ $\zeta_{5}$ $\zeta_{5}^{2}$ $\zeta_{5}^{3}$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ $1$ $5$ $(1,2,3,5,4)$ $\zeta_{5}^{2}$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ $\zeta_{5}$ $\zeta_{5}^{3}$ $1$ $5$ $(1,4,5,3,2)$ $\zeta_{5}^{3}$ $\zeta_{5}$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ $\zeta_{5}^{2}$ $1$ $5$ $(1,3,4,2,5)$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ $\zeta_{5}^{3}$ $\zeta_{5}^{2}$ $\zeta_{5}$
The blue line marks the conjugacy class containing complex conjugation.