# Properties

 Label 4.24048.6t13.a.a Dimension $4$ Group $C_3^2:D_4$ Conductor $24048$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $$24048$$$$\medspace = 2^{4} \cdot 3^{2} \cdot 167$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 6.4.288576.1 Galois orbit size: $1$ Smallest permutation container: $C_3^2:D_4$ Parity: odd Determinant: 1.167.2t1.a.a Projective image: $\SOPlus(4,2)$ Projective stem field: Galois closure of 6.4.288576.1

## Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: $$x^{2} + 7x + 2$$

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

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.