# Properties

 Label 1.21.6t1.a Dimension $1$ Group $C_6$ Conductor $21$ Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $$21$$$$\medspace = 3 \cdot 7$$ Artin number field: Galois closure of 6.0.64827.1 Galois orbit size: $2$ Smallest permutation container: $C_6$ Parity: odd Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $$x^{2} + 24x + 2$$
Roots:
 $r_{ 1 }$ $=$ $$9 a + 12 + \left(16 a + 27\right)\cdot 29 + \left(22 a + 9\right)\cdot 29^{2} + \left(21 a + 22\right)\cdot 29^{3} + 10\cdot 29^{4} +O(29^{5})$$ 9*a + 12 + (16*a + 27)*29 + (22*a + 9)*29^2 + (21*a + 22)*29^3 + 10*29^4+O(29^5) $r_{ 2 }$ $=$ $$18 a + 24 + \left(13 a + 19\right)\cdot 29 + \left(13 a + 8\right)\cdot 29^{2} + \left(10 a + 24\right)\cdot 29^{3} + \left(12 a + 16\right)\cdot 29^{4} +O(29^{5})$$ 18*a + 24 + (13*a + 19)*29 + (13*a + 8)*29^2 + (10*a + 24)*29^3 + (12*a + 16)*29^4+O(29^5) $r_{ 3 }$ $=$ $$13 a + 24 + \left(13 a + 23\right)\cdot 29 + \left(21 a + 18\right)\cdot 29^{2} + \left(21 a + 6\right)\cdot 29^{3} + \left(28 a + 25\right)\cdot 29^{4} +O(29^{5})$$ 13*a + 24 + (13*a + 23)*29 + (21*a + 18)*29^2 + (21*a + 6)*29^3 + (28*a + 25)*29^4+O(29^5) $r_{ 4 }$ $=$ $$20 a + 28 + \left(12 a + 12\right)\cdot 29 + \left(6 a + 19\right)\cdot 29^{2} + \left(7 a + 21\right)\cdot 29^{3} + \left(28 a + 21\right)\cdot 29^{4} +O(29^{5})$$ 20*a + 28 + (12*a + 12)*29 + (6*a + 19)*29^2 + (7*a + 21)*29^3 + (28*a + 21)*29^4+O(29^5) $r_{ 5 }$ $=$ $$11 a + 27 + \left(15 a + 11\right)\cdot 29 + \left(15 a + 4\right)\cdot 29^{2} + \left(18 a + 5\right)\cdot 29^{3} + \left(16 a + 10\right)\cdot 29^{4} +O(29^{5})$$ 11*a + 27 + (15*a + 11)*29 + (15*a + 4)*29^2 + (18*a + 5)*29^3 + (16*a + 10)*29^4+O(29^5) $r_{ 6 }$ $=$ $$16 a + 2 + \left(15 a + 20\right)\cdot 29 + \left(7 a + 25\right)\cdot 29^{2} + \left(7 a + 6\right)\cdot 29^{3} + 2\cdot 29^{4} +O(29^{5})$$ 16*a + 2 + (15*a + 20)*29 + (7*a + 25)*29^2 + (7*a + 6)*29^3 + 2*29^4+O(29^5)

### Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

 Cycle notation $(1,5,6,4,2,3)$ $(1,4)(2,5)(3,6)$

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character values $c1$ $c2$ $1$ $1$ $()$ $1$ $1$ $1$ $2$ $(1,4)(2,5)(3,6)$ $-1$ $-1$ $1$ $3$ $(1,6,2)(3,5,4)$ $\zeta_{3}$ $-\zeta_{3} - 1$ $1$ $3$ $(1,2,6)(3,4,5)$ $-\zeta_{3} - 1$ $\zeta_{3}$ $1$ $6$ $(1,5,6,4,2,3)$ $\zeta_{3} + 1$ $-\zeta_{3}$ $1$ $6$ $(1,3,2,4,6,5)$ $-\zeta_{3}$ $\zeta_{3} + 1$
The blue line marks the conjugacy class containing complex conjugation.