# Properties

 Label 4.4817408.12t34.b.a Dimension $4$ Group $C_3^2:D_4$ Conductor $4817408$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $$4817408$$$$\medspace = 2^{9} \cdot 97^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 6.2.49664.1 Galois orbit size: $1$ Smallest permutation container: 12T34 Parity: even Determinant: 1.8.2t1.a.a Projective image: $\SOPlus(4,2)$ Projective stem field: Galois closure of 6.2.49664.1

## Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $$x^{2} + 45x + 5$$

Roots:
 $r_{ 1 }$ $=$ $$12 + 44\cdot 47 + 40\cdot 47^{2} + 2\cdot 47^{3} + 14\cdot 47^{4} +O(47^{5})$$ 12 + 44*47 + 40*47^2 + 2*47^3 + 14*47^4+O(47^5) $r_{ 2 }$ $=$ $$3 a + 35 + \left(41 a + 3\right)\cdot 47 + \left(21 a + 27\right)\cdot 47^{2} + \left(5 a + 46\right)\cdot 47^{3} + \left(27 a + 40\right)\cdot 47^{4} +O(47^{5})$$ 3*a + 35 + (41*a + 3)*47 + (21*a + 27)*47^2 + (5*a + 46)*47^3 + (27*a + 40)*47^4+O(47^5) $r_{ 3 }$ $=$ $$10 + 19\cdot 47 + 13\cdot 47^{2} + 11\cdot 47^{3} + 9\cdot 47^{4} +O(47^{5})$$ 10 + 19*47 + 13*47^2 + 11*47^3 + 9*47^4+O(47^5) $r_{ 4 }$ $=$ $$44 a + 41 + \left(5 a + 35\right)\cdot 47 + \left(25 a + 29\right)\cdot 47^{2} + \left(41 a + 35\right)\cdot 47^{3} + \left(19 a + 42\right)\cdot 47^{4} +O(47^{5})$$ 44*a + 41 + (5*a + 35)*47 + (25*a + 29)*47^2 + (41*a + 35)*47^3 + (19*a + 42)*47^4+O(47^5) $r_{ 5 }$ $=$ $$17 a + 29 + \left(18 a + 32\right)\cdot 47 + \left(3 a + 20\right)\cdot 47^{2} + \left(34 a + 13\right)\cdot 47^{3} + \left(13 a + 20\right)\cdot 47^{4} +O(47^{5})$$ 17*a + 29 + (18*a + 32)*47 + (3*a + 20)*47^2 + (34*a + 13)*47^3 + (13*a + 20)*47^4+O(47^5) $r_{ 6 }$ $=$ $$30 a + 16 + \left(28 a + 5\right)\cdot 47 + \left(43 a + 9\right)\cdot 47^{2} + \left(12 a + 31\right)\cdot 47^{3} + \left(33 a + 13\right)\cdot 47^{4} +O(47^{5})$$ 30*a + 16 + (28*a + 5)*47 + (43*a + 9)*47^2 + (12*a + 31)*47^3 + (33*a + 13)*47^4+O(47^5)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.