# Properties

 Label 4.3655.8t44.d.a Dimension $4$ Group $C_2 \wr S_4$ Conductor $3655$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $4$ Group: $C_2 \wr S_4$ Conductor: $$3655$$$$\medspace = 5 \cdot 17 \cdot 43$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 8.0.2671805.1 Galois orbit size: $1$ Smallest permutation container: $C_2 \wr S_4$ Parity: odd Determinant: 1.3655.2t1.a.a Projective image: $C_2^3:S_4$ Projective stem field: Galois closure of 8.4.333975625.1

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - x^{5} + x^{4} - x^{3} + 1$$ x^8 - x^5 + x^4 - x^3 + 1 .

The roots of $f$ are computed in an extension of $\Q_{ 167 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 167 }$: $$x^{2} + 166x + 5$$

Roots:
 $r_{ 1 }$ $=$ $$6 a + 151 + \left(158 a + 132\right)\cdot 167 + \left(105 a + 38\right)\cdot 167^{2} + \left(19 a + 8\right)\cdot 167^{3} + \left(154 a + 117\right)\cdot 167^{4} + \left(133 a + 45\right)\cdot 167^{5} + \left(164 a + 38\right)\cdot 167^{6} + \left(99 a + 6\right)\cdot 167^{7} + \left(36 a + 144\right)\cdot 167^{8} + \left(8 a + 74\right)\cdot 167^{9} +O(167^{10})$$ 6*a + 151 + (158*a + 132)*167 + (105*a + 38)*167^2 + (19*a + 8)*167^3 + (154*a + 117)*167^4 + (133*a + 45)*167^5 + (164*a + 38)*167^6 + (99*a + 6)*167^7 + (36*a + 144)*167^8 + (8*a + 74)*167^9+O(167^10) $r_{ 2 }$ $=$ $$105 + 49\cdot 167 + 140\cdot 167^{2} + 35\cdot 167^{3} + 154\cdot 167^{4} + 110\cdot 167^{5} + 97\cdot 167^{6} + 160\cdot 167^{7} + 15\cdot 167^{8} + 89\cdot 167^{9} +O(167^{10})$$ 105 + 49*167 + 140*167^2 + 35*167^3 + 154*167^4 + 110*167^5 + 97*167^6 + 160*167^7 + 15*167^8 + 89*167^9+O(167^10) $r_{ 3 }$ $=$ $$63 a + 25 + \left(102 a + 125\right)\cdot 167 + \left(33 a + 7\right)\cdot 167^{2} + \left(37 a + 147\right)\cdot 167^{3} + \left(56 a + 43\right)\cdot 167^{4} + \left(78 a + 104\right)\cdot 167^{5} + \left(145 a + 96\right)\cdot 167^{6} + \left(72 a + 100\right)\cdot 167^{7} + \left(97 a + 151\right)\cdot 167^{8} + \left(165 a + 143\right)\cdot 167^{9} +O(167^{10})$$ 63*a + 25 + (102*a + 125)*167 + (33*a + 7)*167^2 + (37*a + 147)*167^3 + (56*a + 43)*167^4 + (78*a + 104)*167^5 + (145*a + 96)*167^6 + (72*a + 100)*167^7 + (97*a + 151)*167^8 + (165*a + 143)*167^9+O(167^10) $r_{ 4 }$ $=$ $$a + 53 + \left(40 a + 106\right)\cdot 167 + \left(135 a + 89\right)\cdot 167^{2} + \left(154 a + 9\right)\cdot 167^{3} + \left(84 a + 61\right)\cdot 167^{4} + \left(147 a + 147\right)\cdot 167^{5} + \left(153 a + 137\right)\cdot 167^{6} + \left(93 a + 143\right)\cdot 167^{7} + \left(155 a + 9\right)\cdot 167^{8} + \left(81 a + 3\right)\cdot 167^{9} +O(167^{10})$$ a + 53 + (40*a + 106)*167 + (135*a + 89)*167^2 + (154*a + 9)*167^3 + (84*a + 61)*167^4 + (147*a + 147)*167^5 + (153*a + 137)*167^6 + (93*a + 143)*167^7 + (155*a + 9)*167^8 + (81*a + 3)*167^9+O(167^10) $r_{ 5 }$ $=$ $$161 a + 157 + \left(8 a + 117\right)\cdot 167 + \left(61 a + 153\right)\cdot 167^{2} + \left(147 a + 88\right)\cdot 167^{3} + \left(12 a + 84\right)\cdot 167^{4} + \left(33 a + 25\right)\cdot 167^{5} + \left(2 a + 69\right)\cdot 167^{6} + \left(67 a + 108\right)\cdot 167^{7} + \left(130 a + 80\right)\cdot 167^{8} + \left(158 a + 46\right)\cdot 167^{9} +O(167^{10})$$ 161*a + 157 + (8*a + 117)*167 + (61*a + 153)*167^2 + (147*a + 88)*167^3 + (12*a + 84)*167^4 + (33*a + 25)*167^5 + (2*a + 69)*167^6 + (67*a + 108)*167^7 + (130*a + 80)*167^8 + (158*a + 46)*167^9+O(167^10) $r_{ 6 }$ $=$ $$104 a + 88 + \left(64 a + 164\right)\cdot 167 + \left(133 a + 105\right)\cdot 167^{2} + \left(129 a + 150\right)\cdot 167^{3} + \left(110 a + 62\right)\cdot 167^{4} + \left(88 a + 126\right)\cdot 167^{5} + \left(21 a + 163\right)\cdot 167^{6} + \left(94 a + 27\right)\cdot 167^{7} + \left(69 a + 9\right)\cdot 167^{8} + \left(a + 45\right)\cdot 167^{9} +O(167^{10})$$ 104*a + 88 + (64*a + 164)*167 + (133*a + 105)*167^2 + (129*a + 150)*167^3 + (110*a + 62)*167^4 + (88*a + 126)*167^5 + (21*a + 163)*167^6 + (94*a + 27)*167^7 + (69*a + 9)*167^8 + (a + 45)*167^9+O(167^10) $r_{ 7 }$ $=$ $$35 + 160\cdot 167 + 113\cdot 167^{2} + 31\cdot 167^{3} + 153\cdot 167^{4} + 64\cdot 167^{5} + 87\cdot 167^{6} + 36\cdot 167^{7} + 18\cdot 167^{8} + 2\cdot 167^{9} +O(167^{10})$$ 35 + 160*167 + 113*167^2 + 31*167^3 + 153*167^4 + 64*167^5 + 87*167^6 + 36*167^7 + 18*167^8 + 2*167^9+O(167^10) $r_{ 8 }$ $=$ $$166 a + 54 + \left(126 a + 145\right)\cdot 167 + \left(31 a + 17\right)\cdot 167^{2} + \left(12 a + 29\right)\cdot 167^{3} + \left(82 a + 158\right)\cdot 167^{4} + \left(19 a + 42\right)\cdot 167^{5} + \left(13 a + 144\right)\cdot 167^{6} + \left(73 a + 83\right)\cdot 167^{7} + \left(11 a + 71\right)\cdot 167^{8} + \left(85 a + 96\right)\cdot 167^{9} +O(167^{10})$$ 166*a + 54 + (126*a + 145)*167 + (31*a + 17)*167^2 + (12*a + 29)*167^3 + (82*a + 158)*167^4 + (19*a + 42)*167^5 + (13*a + 144)*167^6 + (73*a + 83)*167^7 + (11*a + 71)*167^8 + (85*a + 96)*167^9+O(167^10)

## Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

 Cycle notation $(1,2,3,4)(5,8,7,6)$ $(1,2)(7,8)$ $(1,8)$

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character value $1$ $1$ $()$ $4$ $1$ $2$ $(1,8)(2,7)(3,6)(4,5)$ $-4$ $4$ $2$ $(3,6)$ $2$ $4$ $2$ $(2,7)(3,6)(4,5)$ $-2$ $6$ $2$ $(1,8)(3,6)$ $0$ $12$ $2$ $(1,3)(2,4)(5,7)(6,8)$ $0$ $12$ $2$ $(1,2)(7,8)$ $2$ $12$ $2$ $(1,4)(2,7)(3,6)(5,8)$ $-2$ $24$ $2$ $(1,2)(3,6)(7,8)$ $0$ $32$ $3$ $(1,3,4)(5,8,6)$ $1$ $12$ $4$ $(1,3,8,6)(2,4,7,5)$ $0$ $12$ $4$ $(1,2,8,7)$ $2$ $12$ $4$ $(1,8)(2,7)(3,5,6,4)$ $-2$ $24$ $4$ $(1,3,8,6)(2,4)(5,7)$ $0$ $24$ $4$ $(1,2,8,7)(3,6)$ $0$ $48$ $4$ $(1,2,3,4)(5,8,7,6)$ $0$ $32$ $6$ $(2,4,3,7,5,6)$ $1$ $32$ $6$ $(1,3,4)(2,7)(5,8,6)$ $-1$ $32$ $6$ $(1,3,5,8,6,4)(2,7)$ $-1$ $48$ $8$ $(1,4,3,7,8,5,6,2)$ $0$

The blue line marks the conjugacy class containing complex conjugation.