# Properties

 Label 4.6642432.12t34.d.a Dimension $4$ Group $C_3^2:D_4$ Conductor $6642432$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $$6642432$$$$\medspace = 2^{8} \cdot 3^{3} \cdot 31^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 6.0.53568.1 Galois orbit size: $1$ Smallest permutation container: 12T34 Parity: odd Determinant: 1.3.2t1.a.a Projective image: $\SOPlus(4,2)$ Projective stem field: Galois closure of 6.0.53568.1

## Defining polynomial

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

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:
 $r_{ 1 }$ $=$ $$28 + 29\cdot 43 + 15\cdot 43^{2} + 16\cdot 43^{3} + 4\cdot 43^{4} +O(43^{5})$$ 28 + 29*43 + 15*43^2 + 16*43^3 + 4*43^4+O(43^5) $r_{ 2 }$ $=$ $$2 a + 27 + \left(29 a + 8\right)\cdot 43 + \left(33 a + 27\right)\cdot 43^{2} + \left(21 a + 13\right)\cdot 43^{3} + \left(38 a + 2\right)\cdot 43^{4} +O(43^{5})$$ 2*a + 27 + (29*a + 8)*43 + (33*a + 27)*43^2 + (21*a + 13)*43^3 + (38*a + 2)*43^4+O(43^5) $r_{ 3 }$ $=$ $$41 a + 9 + \left(39 a + 7\right)\cdot 43 + \left(37 a + 36\right)\cdot 43^{2} + \left(25 a + 40\right)\cdot 43^{3} + \left(33 a + 36\right)\cdot 43^{4} +O(43^{5})$$ 41*a + 9 + (39*a + 7)*43 + (37*a + 36)*43^2 + (25*a + 40)*43^3 + (33*a + 36)*43^4+O(43^5) $r_{ 4 }$ $=$ $$41 a + 29 + \left(13 a + 35\right)\cdot 43 + \left(9 a + 31\right)\cdot 43^{2} + \left(21 a + 1\right)\cdot 43^{3} + \left(4 a + 19\right)\cdot 43^{4} +O(43^{5})$$ 41*a + 29 + (13*a + 35)*43 + (9*a + 31)*43^2 + (21*a + 1)*43^3 + (4*a + 19)*43^4+O(43^5) $r_{ 5 }$ $=$ $$31 + 41\cdot 43 + 26\cdot 43^{2} + 27\cdot 43^{3} + 21\cdot 43^{4} +O(43^{5})$$ 31 + 41*43 + 26*43^2 + 27*43^3 + 21*43^4+O(43^5) $r_{ 6 }$ $=$ $$2 a + 7 + \left(3 a + 6\right)\cdot 43 + \left(5 a + 34\right)\cdot 43^{2} + \left(17 a + 28\right)\cdot 43^{3} + \left(9 a + 1\right)\cdot 43^{4} +O(43^{5})$$ 2*a + 7 + (3*a + 6)*43 + (5*a + 34)*43^2 + (17*a + 28)*43^3 + (9*a + 1)*43^4+O(43^5)

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

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character value $1$ $1$ $()$ $4$ $6$ $2$ $(1,2)(3,4)(5,6)$ $-2$ $6$ $2$ $(3,6)$ $0$ $9$ $2$ $(3,6)(4,5)$ $0$ $4$ $3$ $(1,3,6)$ $-2$ $4$ $3$ $(1,3,6)(2,4,5)$ $1$ $18$ $4$ $(1,2)(3,5,6,4)$ $0$ $12$ $6$ $(1,4,3,5,6,2)$ $1$ $12$ $6$ $(2,4,5)(3,6)$ $0$

The blue line marks the conjugacy class containing complex conjugation.