# Properties

 Label 3.1539.6t6.a.a Dimension $3$ Group $A_4\times C_2$ Conductor $1539$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $3$ Group: $A_4\times C_2$ Conductor: $$1539$$$$\medspace = 3^{4} \cdot 19$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 6.4.124659.1 Galois orbit size: $1$ Smallest permutation container: $A_4\times C_2$ Parity: odd Determinant: 1.19.2t1.a.a Projective image: $A_4$ Projective stem field: Galois closure of 4.0.29241.1

## Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 6.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $$x^{2} + 16x + 3$$

Roots:
 $r_{ 1 }$ $=$ $$12 a + 6 + 15 a\cdot 17 + \left(4 a + 11\right)\cdot 17^{2} + \left(9 a + 10\right)\cdot 17^{3} + \left(13 a + 16\right)\cdot 17^{4} + \left(15 a + 6\right)\cdot 17^{5} +O(17^{6})$$ 12*a + 6 + 15*a*17 + (4*a + 11)*17^2 + (9*a + 10)*17^3 + (13*a + 16)*17^4 + (15*a + 6)*17^5+O(17^6) $r_{ 2 }$ $=$ $$5 a + 1 + \left(a + 4\right)\cdot 17 + 12 a\cdot 17^{2} + \left(7 a + 15\right)\cdot 17^{3} + \left(3 a + 3\right)\cdot 17^{4} + \left(a + 9\right)\cdot 17^{5} +O(17^{6})$$ 5*a + 1 + (a + 4)*17 + 12*a*17^2 + (7*a + 15)*17^3 + (3*a + 3)*17^4 + (a + 9)*17^5+O(17^6) $r_{ 3 }$ $=$ $$13 a + \left(14 a + 3\right)\cdot 17 + \left(4 a + 15\right)\cdot 17^{2} + \left(7 a + 1\right)\cdot 17^{3} + \left(16 a + 13\right)\cdot 17^{4} + \left(3 a + 12\right)\cdot 17^{5} +O(17^{6})$$ 13*a + (14*a + 3)*17 + (4*a + 15)*17^2 + (7*a + 1)*17^3 + (16*a + 13)*17^4 + (3*a + 12)*17^5+O(17^6) $r_{ 4 }$ $=$ $$4 a + 13 + \left(2 a + 4\right)\cdot 17 + \left(12 a + 5\right)\cdot 17^{2} + \left(9 a + 4\right)\cdot 17^{3} + 5\cdot 17^{4} + 13 a\cdot 17^{5} +O(17^{6})$$ 4*a + 13 + (2*a + 4)*17 + (12*a + 5)*17^2 + (9*a + 4)*17^3 + 5*17^4 + 13*a*17^5+O(17^6) $r_{ 5 }$ $=$ $$12 + 13\cdot 17 + 17^{2} + 3\cdot 17^{3} + 5\cdot 17^{4} + 10\cdot 17^{5} +O(17^{6})$$ 12 + 13*17 + 17^2 + 3*17^3 + 5*17^4 + 10*17^5+O(17^6) $r_{ 6 }$ $=$ $$2 + 8\cdot 17 + 16\cdot 17^{3} + 6\cdot 17^{4} + 11\cdot 17^{5} +O(17^{6})$$ 2 + 8*17 + 16*17^3 + 6*17^4 + 11*17^5+O(17^6)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.