# Properties

 Label 1.19.6t1.a.b Dimension $1$ Group $C_6$ Conductor $19$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $$19$$ Artin field: Galois closure of 6.0.2476099.1 Galois orbit size: $2$ Smallest permutation container: $C_6$ Parity: odd Dirichlet character: $$\chi_{19}(8,\cdot)$$ Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - x^{5} + 2x^{4} + 8x^{3} - x^{2} - 5x + 7$$ x^6 - x^5 + 2*x^4 + 8*x^3 - x^2 - 5*x + 7 .

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $$x^{2} + 29x + 3$$

Roots:
 $r_{ 1 }$ $=$ $$17 a + 22 + \left(15 a + 9\right)\cdot 31 + \left(27 a + 14\right)\cdot 31^{2} + \left(15 a + 29\right)\cdot 31^{3} + \left(23 a + 19\right)\cdot 31^{4} +O(31^{5})$$ 17*a + 22 + (15*a + 9)*31 + (27*a + 14)*31^2 + (15*a + 29)*31^3 + (23*a + 19)*31^4+O(31^5) $r_{ 2 }$ $=$ $$14 a + 25 + \left(15 a + 23\right)\cdot 31 + \left(3 a + 22\right)\cdot 31^{2} + \left(15 a + 2\right)\cdot 31^{3} + \left(7 a + 20\right)\cdot 31^{4} +O(31^{5})$$ 14*a + 25 + (15*a + 23)*31 + (3*a + 22)*31^2 + (15*a + 2)*31^3 + (7*a + 20)*31^4+O(31^5) $r_{ 3 }$ $=$ $$10 a + 23 + \left(27 a + 13\right)\cdot 31 + \left(8 a + 26\right)\cdot 31^{2} + \left(15 a + 4\right)\cdot 31^{3} + \left(18 a + 30\right)\cdot 31^{4} +O(31^{5})$$ 10*a + 23 + (27*a + 13)*31 + (8*a + 26)*31^2 + (15*a + 4)*31^3 + (18*a + 30)*31^4+O(31^5) $r_{ 4 }$ $=$ $$21 a + 12 + \left(3 a + 27\right)\cdot 31 + \left(22 a + 16\right)\cdot 31^{2} + \left(15 a + 26\right)\cdot 31^{3} + \left(12 a + 20\right)\cdot 31^{4} +O(31^{5})$$ 21*a + 12 + (3*a + 27)*31 + (22*a + 16)*31^2 + (15*a + 26)*31^3 + (12*a + 20)*31^4+O(31^5) $r_{ 5 }$ $=$ $$11 a + 26 + 29\cdot 31 + \left(23 a + 29\right)\cdot 31^{2} + \left(17 a + 23\right)\cdot 31^{3} + \left(30 a + 25\right)\cdot 31^{4} +O(31^{5})$$ 11*a + 26 + 29*31 + (23*a + 29)*31^2 + (17*a + 23)*31^3 + (30*a + 25)*31^4+O(31^5) $r_{ 6 }$ $=$ $$20 a + 17 + \left(30 a + 19\right)\cdot 31 + \left(7 a + 13\right)\cdot 31^{2} + \left(13 a + 5\right)\cdot 31^{3} + 7\cdot 31^{4} +O(31^{5})$$ 20*a + 17 + (30*a + 19)*31 + (7*a + 13)*31^2 + (13*a + 5)*31^3 + 7*31^4+O(31^5)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.