# Properties

 Label 1.168.6t1.d.b Dimension $1$ Group $C_6$ Conductor $168$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $$168$$$$\medspace = 2^{3} \cdot 3 \cdot 7$$ Artin field: Galois closure of 6.6.232339968.1 Galois orbit size: $2$ Smallest permutation container: $C_6$ Parity: even Dirichlet character: $$\chi_{168}(5,\cdot)$$ Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - 42x^{4} + 504x^{2} - 1512$$ x^6 - 42*x^4 + 504*x^2 - 1512 .

The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $$x^{2} + 42x + 3$$

Roots:
 $r_{ 1 }$ $=$ $$27 a + 8 + \left(12 a + 7\right)\cdot 43 + \left(15 a + 20\right)\cdot 43^{2} + \left(40 a + 30\right)\cdot 43^{3} + \left(28 a + 5\right)\cdot 43^{4} +O(43^{5})$$ 27*a + 8 + (12*a + 7)*43 + (15*a + 20)*43^2 + (40*a + 30)*43^3 + (28*a + 5)*43^4+O(43^5) $r_{ 2 }$ $=$ $$19 a + 12 + \left(42 a + 31\right)\cdot 43 + 20\cdot 43^{2} + \left(30 a + 28\right)\cdot 43^{3} + \left(27 a + 22\right)\cdot 43^{4} +O(43^{5})$$ 19*a + 12 + (42*a + 31)*43 + 20*43^2 + (30*a + 28)*43^3 + (27*a + 22)*43^4+O(43^5) $r_{ 3 }$ $=$ $$42 a + 22 + \left(31 a + 26\right)\cdot 43 + \left(7 a + 33\right)\cdot 43^{2} + \left(31 a + 9\right)\cdot 43^{3} + \left(8 a + 11\right)\cdot 43^{4} +O(43^{5})$$ 42*a + 22 + (31*a + 26)*43 + (7*a + 33)*43^2 + (31*a + 9)*43^3 + (8*a + 11)*43^4+O(43^5) $r_{ 4 }$ $=$ $$16 a + 35 + \left(30 a + 35\right)\cdot 43 + \left(27 a + 22\right)\cdot 43^{2} + \left(2 a + 12\right)\cdot 43^{3} + \left(14 a + 37\right)\cdot 43^{4} +O(43^{5})$$ 16*a + 35 + (30*a + 35)*43 + (27*a + 22)*43^2 + (2*a + 12)*43^3 + (14*a + 37)*43^4+O(43^5) $r_{ 5 }$ $=$ $$24 a + 31 + 11\cdot 43 + \left(42 a + 22\right)\cdot 43^{2} + \left(12 a + 14\right)\cdot 43^{3} + \left(15 a + 20\right)\cdot 43^{4} +O(43^{5})$$ 24*a + 31 + 11*43 + (42*a + 22)*43^2 + (12*a + 14)*43^3 + (15*a + 20)*43^4+O(43^5) $r_{ 6 }$ $=$ $$a + 21 + \left(11 a + 16\right)\cdot 43 + \left(35 a + 9\right)\cdot 43^{2} + \left(11 a + 33\right)\cdot 43^{3} + \left(34 a + 31\right)\cdot 43^{4} +O(43^{5})$$ a + 21 + (11*a + 16)*43 + (35*a + 9)*43^2 + (11*a + 33)*43^3 + (34*a + 31)*43^4+O(43^5)

## 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 value $1$ $1$ $()$ $1$ $1$ $2$ $(1,4)(2,5)(3,6)$ $-1$ $1$ $3$ $(1,6,2)(3,5,4)$ $-\zeta_{3} - 1$ $1$ $3$ $(1,2,6)(3,4,5)$ $\zeta_{3}$ $1$ $6$ $(1,5,6,4,2,3)$ $-\zeta_{3}$ $1$ $6$ $(1,3,2,4,6,5)$ $\zeta_{3} + 1$

The blue line marks the conjugacy class containing complex conjugation.