# Properties

 Label 18.417...568.36t1758.a.a Dimension $18$ Group $S_4\wr C_2$ Conductor $4.173\times 10^{30}$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $18$ Group: $S_4\wr C_2$ Conductor: $$417\!\cdots\!568$$$$\medspace = 2^{51} \cdot 3^{32}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 8.2.1719926784.1 Galois orbit size: $1$ Smallest permutation container: 36T1758 Parity: odd Determinant: 1.8.2t1.b.a Projective image: $S_4\wr C_2$ Projective stem field: Galois closure of 8.2.1719926784.1

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - 4x^{7} + 8x^{6} - 8x^{5} - x^{4} + 20x^{3} - 28x^{2} + 16x - 2$$ x^8 - 4*x^7 + 8*x^6 - 8*x^5 - x^4 + 20*x^3 - 28*x^2 + 16*x - 2 .

The roots of $f$ are computed in an extension of $\Q_{ 103 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 103 }$: $$x^{2} + 102x + 5$$

Roots:
 $r_{ 1 }$ $=$ $$75 a + 18 + \left(22 a + 74\right)\cdot 103 + \left(55 a + 7\right)\cdot 103^{2} + \left(58 a + 50\right)\cdot 103^{3} + \left(84 a + 61\right)\cdot 103^{4} + \left(94 a + 28\right)\cdot 103^{5} + \left(84 a + 1\right)\cdot 103^{6} + \left(18 a + 43\right)\cdot 103^{7} + \left(15 a + 73\right)\cdot 103^{8} + \left(3 a + 89\right)\cdot 103^{9} +O(103^{10})$$ 75*a + 18 + (22*a + 74)*103 + (55*a + 7)*103^2 + (58*a + 50)*103^3 + (84*a + 61)*103^4 + (94*a + 28)*103^5 + (84*a + 1)*103^6 + (18*a + 43)*103^7 + (15*a + 73)*103^8 + (3*a + 89)*103^9+O(103^10) $r_{ 2 }$ $=$ $$83 + 50\cdot 103 + 60\cdot 103^{2} + 67\cdot 103^{3} + 2\cdot 103^{4} + 76\cdot 103^{5} + 44\cdot 103^{6} + 34\cdot 103^{7} + 74\cdot 103^{8} + 38\cdot 103^{9} +O(103^{10})$$ 83 + 50*103 + 60*103^2 + 67*103^3 + 2*103^4 + 76*103^5 + 44*103^6 + 34*103^7 + 74*103^8 + 38*103^9+O(103^10) $r_{ 3 }$ $=$ $$39 a + 1 + \left(95 a + 1\right)\cdot 103 + \left(22 a + 2\right)\cdot 103^{2} + \left(48 a + 17\right)\cdot 103^{3} + \left(67 a + 26\right)\cdot 103^{4} + \left(16 a + 12\right)\cdot 103^{5} + \left(16 a + 55\right)\cdot 103^{6} + \left(82 a + 20\right)\cdot 103^{7} + \left(47 a + 102\right)\cdot 103^{8} + \left(46 a + 40\right)\cdot 103^{9} +O(103^{10})$$ 39*a + 1 + (95*a + 1)*103 + (22*a + 2)*103^2 + (48*a + 17)*103^3 + (67*a + 26)*103^4 + (16*a + 12)*103^5 + (16*a + 55)*103^6 + (82*a + 20)*103^7 + (47*a + 102)*103^8 + (46*a + 40)*103^9+O(103^10) $r_{ 4 }$ $=$ $$11 a + 24 + \left(35 a + 6\right)\cdot 103 + \left(30 a + 29\right)\cdot 103^{2} + \left(13 a + 52\right)\cdot 103^{3} + \left(91 a + 55\right)\cdot 103^{4} + \left(43 a + 54\right)\cdot 103^{5} + \left(70 a + 3\right)\cdot 103^{6} + \left(29 a + 76\right)\cdot 103^{7} + \left(84 a + 86\right)\cdot 103^{8} + \left(44 a + 98\right)\cdot 103^{9} +O(103^{10})$$ 11*a + 24 + (35*a + 6)*103 + (30*a + 29)*103^2 + (13*a + 52)*103^3 + (91*a + 55)*103^4 + (43*a + 54)*103^5 + (70*a + 3)*103^6 + (29*a + 76)*103^7 + (84*a + 86)*103^8 + (44*a + 98)*103^9+O(103^10) $r_{ 5 }$ $=$ $$92 a + 35 + \left(67 a + 30\right)\cdot 103 + \left(72 a + 24\right)\cdot 103^{2} + \left(89 a + 35\right)\cdot 103^{3} + \left(11 a + 30\right)\cdot 103^{4} + \left(59 a + 7\right)\cdot 103^{5} + \left(32 a + 30\right)\cdot 103^{6} + \left(73 a + 35\right)\cdot 103^{7} + \left(18 a + 38\right)\cdot 103^{8} + \left(58 a + 59\right)\cdot 103^{9} +O(103^{10})$$ 92*a + 35 + (67*a + 30)*103 + (72*a + 24)*103^2 + (89*a + 35)*103^3 + (11*a + 30)*103^4 + (59*a + 7)*103^5 + (32*a + 30)*103^6 + (73*a + 35)*103^7 + (18*a + 38)*103^8 + (58*a + 59)*103^9+O(103^10) $r_{ 6 }$ $=$ $$19 + 67\cdot 103 + 9\cdot 103^{2} + 94\cdot 103^{3} + 102\cdot 103^{4} + 26\cdot 103^{5} + 25\cdot 103^{6} + 36\cdot 103^{7} + 2\cdot 103^{8} + 70\cdot 103^{9} +O(103^{10})$$ 19 + 67*103 + 9*103^2 + 94*103^3 + 102*103^4 + 26*103^5 + 25*103^6 + 36*103^7 + 2*103^8 + 70*103^9+O(103^10) $r_{ 7 }$ $=$ $$64 a + 40 + \left(7 a + 57\right)\cdot 103 + \left(80 a + 32\right)\cdot 103^{2} + \left(54 a + 42\right)\cdot 103^{3} + \left(35 a + 45\right)\cdot 103^{4} + \left(86 a + 64\right)\cdot 103^{5} + \left(86 a + 54\right)\cdot 103^{6} + \left(20 a + 86\right)\cdot 103^{7} + \left(55 a + 67\right)\cdot 103^{8} + \left(56 a + 39\right)\cdot 103^{9} +O(103^{10})$$ 64*a + 40 + (7*a + 57)*103 + (80*a + 32)*103^2 + (54*a + 42)*103^3 + (35*a + 45)*103^4 + (86*a + 64)*103^5 + (86*a + 54)*103^6 + (20*a + 86)*103^7 + (55*a + 67)*103^8 + (56*a + 39)*103^9+O(103^10) $r_{ 8 }$ $=$ $$28 a + 93 + \left(80 a + 21\right)\cdot 103 + \left(47 a + 40\right)\cdot 103^{2} + \left(44 a + 53\right)\cdot 103^{3} + \left(18 a + 87\right)\cdot 103^{4} + \left(8 a + 38\right)\cdot 103^{5} + \left(18 a + 94\right)\cdot 103^{6} + \left(84 a + 79\right)\cdot 103^{7} + \left(87 a + 69\right)\cdot 103^{8} + \left(99 a + 77\right)\cdot 103^{9} +O(103^{10})$$ 28*a + 93 + (80*a + 21)*103 + (47*a + 40)*103^2 + (44*a + 53)*103^3 + (18*a + 87)*103^4 + (8*a + 38)*103^5 + (18*a + 94)*103^6 + (84*a + 79)*103^7 + (87*a + 69)*103^8 + (99*a + 77)*103^9+O(103^10)

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

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character value $1$ $1$ $()$ $18$ $6$ $2$ $(1,5)(4,8)$ $-6$ $9$ $2$ $(1,5)(2,6)(3,7)(4,8)$ $2$ $12$ $2$ $(2,3)$ $0$ $24$ $2$ $(1,2)(3,4)(5,6)(7,8)$ $0$ $36$ $2$ $(1,4)(2,3)$ $-2$ $36$ $2$ $(1,5)(2,3)(4,8)$ $0$ $16$ $3$ $(2,6,7)$ $0$ $64$ $3$ $(2,6,7)(4,5,8)$ $0$ $12$ $4$ $(1,4,5,8)$ $0$ $36$ $4$ $(1,4,5,8)(2,3,6,7)$ $-2$ $36$ $4$ $(1,5)(2,3,6,7)(4,8)$ $0$ $72$ $4$ $(1,6,5,2)(3,4,7,8)$ $0$ $72$ $4$ $(1,4,5,8)(2,3)$ $2$ $144$ $4$ $(1,2,4,3)(5,6)(7,8)$ $0$ $48$ $6$ $(1,5)(2,7,6)(4,8)$ $0$ $96$ $6$ $(2,3)(4,8,5)$ $0$ $192$ $6$ $(1,3)(2,4,6,5,7,8)$ $0$ $144$ $8$ $(1,3,4,6,5,7,8,2)$ $0$ $96$ $12$ $(1,4,5,8)(2,6,7)$ $0$

The blue line marks the conjugacy class containing complex conjugation.