# Properties

 Label 2.168.6t5.c.a Dimension $2$ Group $S_3\times C_3$ Conductor $168$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3\times C_3$ Conductor: $$168$$$$\medspace = 2^{3} \cdot 3 \cdot 7$$ Artin stem field: Galois closure of 6.0.677376.1 Galois orbit size: $2$ Smallest permutation container: $S_3\times C_3$ Parity: odd Determinant: 1.168.6t1.c.a Projective image: $S_3$ Projective stem field: Galois closure of 3.1.1176.1

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 8x^{2} - 4x + 1$$ x^6 - 2*x^5 + 3*x^4 - 6*x^3 + 8*x^2 - 4*x + 1 .

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 6.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $$x^{2} + 12x + 2$$

Roots:
 $r_{ 1 }$ $=$ $$a + 11 + \left(7 a + 7\right)\cdot 13 + 7\cdot 13^{2} + \left(9 a + 7\right)\cdot 13^{3} + 12 a\cdot 13^{4} + \left(10 a + 8\right)\cdot 13^{5} +O(13^{6})$$ a + 11 + (7*a + 7)*13 + 7*13^2 + (9*a + 7)*13^3 + 12*a*13^4 + (10*a + 8)*13^5+O(13^6) $r_{ 2 }$ $=$ $$12 a + 12 + 5 a\cdot 13 + \left(12 a + 1\right)\cdot 13^{2} + \left(3 a + 3\right)\cdot 13^{3} + 4\cdot 13^{4} + \left(2 a + 6\right)\cdot 13^{5} +O(13^{6})$$ 12*a + 12 + 5*a*13 + (12*a + 1)*13^2 + (3*a + 3)*13^3 + 4*13^4 + (2*a + 6)*13^5+O(13^6) $r_{ 3 }$ $=$ $$10 a + 11 + \left(8 a + 6\right)\cdot 13 + \left(10 a + 9\right)\cdot 13^{2} + 2\cdot 13^{3} + \left(12 a + 7\right)\cdot 13^{4} + \left(3 a + 1\right)\cdot 13^{5} +O(13^{6})$$ 10*a + 11 + (8*a + 6)*13 + (10*a + 9)*13^2 + 2*13^3 + (12*a + 7)*13^4 + (3*a + 1)*13^5+O(13^6) $r_{ 4 }$ $=$ $$5 a + 10 + \left(3 a + 9\right)\cdot 13 + \left(7 a + 2\right)\cdot 13^{2} + \left(12 a + 7\right)\cdot 13^{3} + \left(7 a + 6\right)\cdot 13^{4} + \left(3 a + 10\right)\cdot 13^{5} +O(13^{6})$$ 5*a + 10 + (3*a + 9)*13 + (7*a + 2)*13^2 + (12*a + 7)*13^3 + (7*a + 6)*13^4 + (3*a + 10)*13^5+O(13^6) $r_{ 5 }$ $=$ $$8 a + 2 + \left(9 a + 8\right)\cdot 13 + \left(5 a + 6\right)\cdot 13^{2} + 12\cdot 13^{3} + \left(5 a + 1\right)\cdot 13^{4} + \left(9 a + 6\right)\cdot 13^{5} +O(13^{6})$$ 8*a + 2 + (9*a + 8)*13 + (5*a + 6)*13^2 + 12*13^3 + (5*a + 1)*13^4 + (9*a + 6)*13^5+O(13^6) $r_{ 6 }$ $=$ $$3 a + 8 + \left(4 a + 5\right)\cdot 13 + \left(2 a + 11\right)\cdot 13^{2} + \left(12 a + 5\right)\cdot 13^{3} + 5\cdot 13^{4} + \left(9 a + 6\right)\cdot 13^{5} +O(13^{6})$$ 3*a + 8 + (4*a + 5)*13 + (2*a + 11)*13^2 + (12*a + 5)*13^3 + 5*13^4 + (9*a + 6)*13^5+O(13^6)

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

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character value $1$ $1$ $()$ $2$ $3$ $2$ $(1,6)(2,4)(3,5)$ $0$ $1$ $3$ $(1,3,4)(2,6,5)$ $2 \zeta_{3}$ $1$ $3$ $(1,4,3)(2,5,6)$ $-2 \zeta_{3} - 2$ $2$ $3$ $(1,3,4)$ $\zeta_{3} + 1$ $2$ $3$ $(1,4,3)$ $-\zeta_{3}$ $2$ $3$ $(1,4,3)(2,6,5)$ $-1$ $3$ $6$ $(1,2,3,6,4,5)$ $0$ $3$ $6$ $(1,5,4,6,3,2)$ $0$

The blue line marks the conjugacy class containing complex conjugation.