# Properties

 Label 1.171.6t1.c.a Dimension $1$ Group $C_6$ Conductor $171$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $$171$$$$\medspace = 3^{2} \cdot 19$$ Artin field: Galois closure of 6.0.16245685539.1 Galois orbit size: $2$ Smallest permutation container: $C_6$ Parity: odd Dirichlet character: $$\chi_{171}(160,\cdot)$$ Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - 95x^{3} + 684x^{2} - 570x + 2375$$ x^6 - 95*x^3 + 684*x^2 - 570*x + 2375 .

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:
 $r_{ 1 }$ $=$ $$a + 14 + \left(11 a + 27\right)\cdot 29 + \left(14 a + 19\right)\cdot 29^{2} + \left(15 a + 27\right)\cdot 29^{3} + \left(7 a + 11\right)\cdot 29^{4} +O(29^{5})$$ a + 14 + (11*a + 27)*29 + (14*a + 19)*29^2 + (15*a + 27)*29^3 + (7*a + 11)*29^4+O(29^5) $r_{ 2 }$ $=$ $$12 a + \left(4 a + 23\right)\cdot 29 + \left(18 a + 23\right)\cdot 29^{2} + \left(19 a + 2\right)\cdot 29^{3} + \left(15 a + 11\right)\cdot 29^{4} +O(29^{5})$$ 12*a + (4*a + 23)*29 + (18*a + 23)*29^2 + (19*a + 2)*29^3 + (15*a + 11)*29^4+O(29^5) $r_{ 3 }$ $=$ $$17 a + 2 + \left(24 a + 4\right)\cdot 29 + \left(10 a + 23\right)\cdot 29^{2} + \left(9 a + 24\right)\cdot 29^{3} + \left(13 a + 11\right)\cdot 29^{4} +O(29^{5})$$ 17*a + 2 + (24*a + 4)*29 + (10*a + 23)*29^2 + (9*a + 24)*29^3 + (13*a + 11)*29^4+O(29^5) $r_{ 4 }$ $=$ $$28 a + 19 + \left(17 a + 23\right)\cdot 29 + \left(14 a + 22\right)\cdot 29^{2} + \left(13 a + 3\right)\cdot 29^{3} + \left(21 a + 5\right)\cdot 29^{4} +O(29^{5})$$ 28*a + 19 + (17*a + 23)*29 + (14*a + 22)*29^2 + (13*a + 3)*29^3 + (21*a + 5)*29^4+O(29^5) $r_{ 5 }$ $=$ $$13 a + 8 + \left(15 a + 1\right)\cdot 29 + \left(3 a + 12\right)\cdot 29^{2} + 6 a\cdot 29^{3} + \left(23 a + 12\right)\cdot 29^{4} +O(29^{5})$$ 13*a + 8 + (15*a + 1)*29 + (3*a + 12)*29^2 + 6*a*29^3 + (23*a + 12)*29^4+O(29^5) $r_{ 6 }$ $=$ $$16 a + 15 + \left(13 a + 7\right)\cdot 29 + \left(25 a + 14\right)\cdot 29^{2} + \left(22 a + 27\right)\cdot 29^{3} + \left(5 a + 5\right)\cdot 29^{4} +O(29^{5})$$ 16*a + 15 + (13*a + 7)*29 + (25*a + 14)*29^2 + (22*a + 27)*29^3 + (5*a + 5)*29^4+O(29^5)

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

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character value $1$ $1$ $()$ $1$ $1$ $2$ $(1,4)(2,3)(5,6)$ $-1$ $1$ $3$ $(1,2,6)(3,5,4)$ $-\zeta_{3} - 1$ $1$ $3$ $(1,6,2)(3,4,5)$ $\zeta_{3}$ $1$ $6$ $(1,3,6,4,2,5)$ $\zeta_{3} + 1$ $1$ $6$ $(1,5,2,4,6,3)$ $-\zeta_{3}$

The blue line marks the conjugacy class containing complex conjugation.