# Properties

 Label 4.6642432.12t34.c.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 6.4.91517952.1 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 6.4.91517952.1

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - 4x^{4} - 16x^{3} - 27x^{2} + 32x + 64$$ x^6 - 4*x^4 - 16*x^3 - 27*x^2 + 32*x + 64 .

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:
 $r_{ 1 }$ $=$ $$15 a + 27 + \left(3 a + 36\right)\cdot 43 + \left(22 a + 8\right)\cdot 43^{2} + \left(14 a + 39\right)\cdot 43^{3} + \left(9 a + 38\right)\cdot 43^{4} +O(43^{5})$$ 15*a + 27 + (3*a + 36)*43 + (22*a + 8)*43^2 + (14*a + 39)*43^3 + (9*a + 38)*43^4+O(43^5) $r_{ 2 }$ $=$ $$16 + 18\cdot 43 + 14\cdot 43^{2} + 6\cdot 43^{3} + 2\cdot 43^{4} +O(43^{5})$$ 16 + 18*43 + 14*43^2 + 6*43^3 + 2*43^4+O(43^5) $r_{ 3 }$ $=$ $$39 a + 37 + \left(35 a + 13\right)\cdot 43 + \left(9 a + 27\right)\cdot 43^{2} + \left(37 a + 4\right)\cdot 43^{3} + \left(24 a + 5\right)\cdot 43^{4} +O(43^{5})$$ 39*a + 37 + (35*a + 13)*43 + (9*a + 27)*43^2 + (37*a + 4)*43^3 + (24*a + 5)*43^4+O(43^5) $r_{ 4 }$ $=$ $$4 a + 33 + \left(7 a + 10\right)\cdot 43 + \left(33 a + 1\right)\cdot 43^{2} + \left(5 a + 32\right)\cdot 43^{3} + \left(18 a + 35\right)\cdot 43^{4} +O(43^{5})$$ 4*a + 33 + (7*a + 10)*43 + (33*a + 1)*43^2 + (5*a + 32)*43^3 + (18*a + 35)*43^4+O(43^5) $r_{ 5 }$ $=$ $$17 + 24\cdot 43 + 6\cdot 43^{2} + 15\cdot 43^{3} + 13\cdot 43^{4} +O(43^{5})$$ 17 + 24*43 + 6*43^2 + 15*43^3 + 13*43^4+O(43^5) $r_{ 6 }$ $=$ $$28 a + 42 + \left(39 a + 24\right)\cdot 43 + \left(20 a + 27\right)\cdot 43^{2} + \left(28 a + 31\right)\cdot 43^{3} + \left(33 a + 33\right)\cdot 43^{4} +O(43^{5})$$ 28*a + 42 + (39*a + 24)*43 + (20*a + 27)*43^2 + (28*a + 31)*43^3 + (33*a + 33)*43^4+O(43^5)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.