# Properties

 Label 4.40428032.12t34.b.a Dimension $4$ Group $C_3^2:D_4$ Conductor $40428032$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $$40428032$$$$\medspace = 2^{9} \cdot 281^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 6.0.143872.1 Galois orbit size: $1$ Smallest permutation container: 12T34 Parity: odd Determinant: 1.8.2t1.b.a Projective image: $\SOPlus(4,2)$ Projective stem field: Galois closure of 6.0.143872.1

## 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 $\Q_{ 17 }$ to precision 5.

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

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

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.