# Properties

 Label 1.171.6t1.a Dimension $1$ Group $C_6$ Conductor $171$ Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $$171$$$$\medspace = 3^{2} \cdot 19$$ Artin number field: Galois closure of 6.6.135005697.1 Galois orbit size: $2$ Smallest permutation container: $C_6$ 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 an extension of $\Q_{ 17 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $$x^{2} + 16x + 3$$
Roots:
 $r_{ 1 }$ $=$ $$2 a + 6 + \left(2 a + 2\right)\cdot 17 + \left(5 a + 8\right)\cdot 17^{2} + \left(16 a + 12\right)\cdot 17^{3} + \left(11 a + 16\right)\cdot 17^{4} +O(17^{5})$$ 2*a + 6 + (2*a + 2)*17 + (5*a + 8)*17^2 + (16*a + 12)*17^3 + (11*a + 16)*17^4+O(17^5) $r_{ 2 }$ $=$ $$8 a + 11 + \left(15 a + 8\right)\cdot 17 + \left(5 a + 6\right)\cdot 17^{2} + \left(14 a + 7\right)\cdot 17^{3} + \left(12 a + 1\right)\cdot 17^{4} +O(17^{5})$$ 8*a + 11 + (15*a + 8)*17 + (5*a + 6)*17^2 + (14*a + 7)*17^3 + (12*a + 1)*17^4+O(17^5) $r_{ 3 }$ $=$ $$6 a + 9 + \left(13 a + 15\right)\cdot 17 + 11\cdot 17^{2} + \left(15 a + 5\right)\cdot 17^{3} +O(17^{5})$$ 6*a + 9 + (13*a + 15)*17 + 11*17^2 + (15*a + 5)*17^3+O(17^5) $r_{ 4 }$ $=$ $$15 a + 8 + \left(14 a + 2\right)\cdot 17 + \left(11 a + 11\right)\cdot 17^{2} + 6\cdot 17^{3} + \left(5 a + 12\right)\cdot 17^{4} +O(17^{5})$$ 15*a + 8 + (14*a + 2)*17 + (11*a + 11)*17^2 + 6*17^3 + (5*a + 12)*17^4+O(17^5) $r_{ 5 }$ $=$ $$11 a + 15 + \left(3 a + 5\right)\cdot 17 + \left(16 a + 16\right)\cdot 17^{2} + \left(a + 2\right)\cdot 17^{3} + \left(16 a + 3\right)\cdot 17^{4} +O(17^{5})$$ 11*a + 15 + (3*a + 5)*17 + (16*a + 16)*17^2 + (a + 2)*17^3 + (16*a + 3)*17^4+O(17^5) $r_{ 6 }$ $=$ $$9 a + 2 + \left(a + 16\right)\cdot 17 + \left(11 a + 13\right)\cdot 17^{2} + \left(2 a + 15\right)\cdot 17^{3} + \left(4 a + 16\right)\cdot 17^{4} +O(17^{5})$$ 9*a + 2 + (a + 16)*17 + (11*a + 13)*17^2 + (2*a + 15)*17^3 + (4*a + 16)*17^4+O(17^5)

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

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

### Character values on conjugacy classes

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