# Properties

 Label 2.84.6t5.b.a Dimension $2$ Group $S_3\times C_3$ Conductor $84$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3\times C_3$ Conductor: $$84$$$$\medspace = 2^{2} \cdot 3 \cdot 7$$ Artin stem field: Galois closure of 6.0.21168.1 Galois orbit size: $2$ Smallest permutation container: $S_3\times C_3$ Parity: odd Determinant: 1.21.6t1.a.b Projective image: $S_3$ Projective stem field: Galois closure of 3.1.588.1

## Defining polynomial

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

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:
 $r_{ 1 }$ $=$ $$21 a + 17 + \left(8 a + 17\right)\cdot 29 + \left(25 a + 14\right)\cdot 29^{2} + \left(24 a + 20\right)\cdot 29^{3} + \left(3 a + 3\right)\cdot 29^{4} +O(29^{5})$$ 21*a + 17 + (8*a + 17)*29 + (25*a + 14)*29^2 + (24*a + 20)*29^3 + (3*a + 3)*29^4+O(29^5) $r_{ 2 }$ $=$ $$12 a + 28 + \left(27 a + 15\right)\cdot 29 + \left(11 a + 12\right)\cdot 29^{2} + \left(20 a + 19\right)\cdot 29^{3} + \left(21 a + 22\right)\cdot 29^{4} +O(29^{5})$$ 12*a + 28 + (27*a + 15)*29 + (11*a + 12)*29^2 + (20*a + 19)*29^3 + (21*a + 22)*29^4+O(29^5) $r_{ 3 }$ $=$ $$8 a + 6 + \left(20 a + 11\right)\cdot 29 + \left(3 a + 16\right)\cdot 29^{2} + \left(4 a + 3\right)\cdot 29^{3} + \left(25 a + 27\right)\cdot 29^{4} +O(29^{5})$$ 8*a + 6 + (20*a + 11)*29 + (3*a + 16)*29^2 + (4*a + 3)*29^3 + (25*a + 27)*29^4+O(29^5) $r_{ 4 }$ $=$ $$17 a + 1 + \left(a + 25\right)\cdot 29 + \left(17 a + 15\right)\cdot 29^{2} + \left(8 a + 22\right)\cdot 29^{3} + \left(7 a + 23\right)\cdot 29^{4} +O(29^{5})$$ 17*a + 1 + (a + 25)*29 + (17*a + 15)*29^2 + (8*a + 22)*29^3 + (7*a + 23)*29^4+O(29^5) $r_{ 5 }$ $=$ $$6 a + 4 + \left(2 a + 6\right)\cdot 29 + \left(a + 12\right)\cdot 29^{2} + \left(14 a + 19\right)\cdot 29^{3} + 4 a\cdot 29^{4} +O(29^{5})$$ 6*a + 4 + (2*a + 6)*29 + (a + 12)*29^2 + (14*a + 19)*29^3 + 4*a*29^4+O(29^5) $r_{ 6 }$ $=$ $$23 a + 5 + \left(26 a + 11\right)\cdot 29 + \left(27 a + 15\right)\cdot 29^{2} + \left(14 a + 1\right)\cdot 29^{3} + \left(24 a + 9\right)\cdot 29^{4} +O(29^{5})$$ 23*a + 5 + (26*a + 11)*29 + (27*a + 15)*29^2 + (14*a + 1)*29^3 + (24*a + 9)*29^4+O(29^5)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.