# Properties

 Label 18.114...592.36t1758.a.a Dimension $18$ Group $S_4\wr C_2$ Conductor $1.147\times 10^{31}$ Root number $1$ Indicator $1$

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## Basic invariants

 Dimension: $18$ Group: $S_4\wr C_2$ Conductor: $$114\!\cdots\!592$$$$\medspace = 2^{12} \cdot 3^{14} \cdot 7^{9} \cdot 29^{9}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 8.2.57748079088.1 Galois orbit size: $1$ Smallest permutation container: 36T1758 Parity: odd Determinant: 1.203.2t1.a.a Projective image: $S_4\wr C_2$ Projective stem field: Galois closure of 8.2.57748079088.1

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - x^{7} + 5x^{6} - 4x^{5} + 10x^{4} + 10x^{3} - 4x^{2} + 4x + 4$$ x^8 - x^7 + 5*x^6 - 4*x^5 + 10*x^4 + 10*x^3 - 4*x^2 + 4*x + 4 .

The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $$x^{2} + 58x + 2$$

Roots:
 $r_{ 1 }$ $=$ $$57 a + 55 + \left(15 a + 49\right)\cdot 59 + \left(26 a + 14\right)\cdot 59^{2} + \left(4 a + 56\right)\cdot 59^{3} + \left(8 a + 43\right)\cdot 59^{4} + \left(13 a + 35\right)\cdot 59^{5} + \left(26 a + 10\right)\cdot 59^{6} + \left(49 a + 46\right)\cdot 59^{7} + \left(a + 39\right)\cdot 59^{8} + \left(22 a + 50\right)\cdot 59^{9} +O(59^{10})$$ 57*a + 55 + (15*a + 49)*59 + (26*a + 14)*59^2 + (4*a + 56)*59^3 + (8*a + 43)*59^4 + (13*a + 35)*59^5 + (26*a + 10)*59^6 + (49*a + 46)*59^7 + (a + 39)*59^8 + (22*a + 50)*59^9+O(59^10) $r_{ 2 }$ $=$ $$2 a + 53 + \left(43 a + 8\right)\cdot 59 + \left(32 a + 25\right)\cdot 59^{2} + \left(54 a + 34\right)\cdot 59^{3} + \left(50 a + 47\right)\cdot 59^{4} + \left(45 a + 40\right)\cdot 59^{5} + \left(32 a + 23\right)\cdot 59^{6} + \left(9 a + 10\right)\cdot 59^{7} + \left(57 a + 51\right)\cdot 59^{8} + \left(36 a + 11\right)\cdot 59^{9} +O(59^{10})$$ 2*a + 53 + (43*a + 8)*59 + (32*a + 25)*59^2 + (54*a + 34)*59^3 + (50*a + 47)*59^4 + (45*a + 40)*59^5 + (32*a + 23)*59^6 + (9*a + 10)*59^7 + (57*a + 51)*59^8 + (36*a + 11)*59^9+O(59^10) $r_{ 3 }$ $=$ $$5 a + 13 + \left(46 a + 1\right)\cdot 59 + \left(33 a + 1\right)\cdot 59^{2} + \left(25 a + 33\right)\cdot 59^{3} + \left(37 a + 53\right)\cdot 59^{4} + \left(25 a + 44\right)\cdot 59^{5} + \left(54 a + 40\right)\cdot 59^{6} + \left(45 a + 13\right)\cdot 59^{7} + \left(8 a + 50\right)\cdot 59^{8} + \left(36 a + 17\right)\cdot 59^{9} +O(59^{10})$$ 5*a + 13 + (46*a + 1)*59 + (33*a + 1)*59^2 + (25*a + 33)*59^3 + (37*a + 53)*59^4 + (25*a + 44)*59^5 + (54*a + 40)*59^6 + (45*a + 13)*59^7 + (8*a + 50)*59^8 + (36*a + 17)*59^9+O(59^10) $r_{ 4 }$ $=$ $$31 a + 20 + \left(2 a + 13\right)\cdot 59 + \left(52 a + 15\right)\cdot 59^{2} + \left(11 a + 29\right)\cdot 59^{3} + \left(50 a + 2\right)\cdot 59^{4} + \left(17 a + 57\right)\cdot 59^{5} + 21 a\cdot 59^{6} + \left(24 a + 3\right)\cdot 59^{7} + \left(13 a + 6\right)\cdot 59^{8} + \left(47 a + 49\right)\cdot 59^{9} +O(59^{10})$$ 31*a + 20 + (2*a + 13)*59 + (52*a + 15)*59^2 + (11*a + 29)*59^3 + (50*a + 2)*59^4 + (17*a + 57)*59^5 + 21*a*59^6 + (24*a + 3)*59^7 + (13*a + 6)*59^8 + (47*a + 49)*59^9+O(59^10) $r_{ 5 }$ $=$ $$28 a + 51 + \left(56 a + 43\right)\cdot 59 + \left(6 a + 5\right)\cdot 59^{2} + \left(47 a + 48\right)\cdot 59^{3} + \left(8 a + 40\right)\cdot 59^{4} + \left(41 a + 24\right)\cdot 59^{5} + \left(37 a + 4\right)\cdot 59^{6} + \left(34 a + 6\right)\cdot 59^{7} + \left(45 a + 54\right)\cdot 59^{8} + \left(11 a + 23\right)\cdot 59^{9} +O(59^{10})$$ 28*a + 51 + (56*a + 43)*59 + (6*a + 5)*59^2 + (47*a + 48)*59^3 + (8*a + 40)*59^4 + (41*a + 24)*59^5 + (37*a + 4)*59^6 + (34*a + 6)*59^7 + (45*a + 54)*59^8 + (11*a + 23)*59^9+O(59^10) $r_{ 6 }$ $=$ $$9 + 45\cdot 59 + 34\cdot 59^{2} + 37\cdot 59^{3} + 29\cdot 59^{4} + 20\cdot 59^{5} + 19\cdot 59^{6} + 2\cdot 59^{7} + 49\cdot 59^{8} + 57\cdot 59^{9} +O(59^{10})$$ 9 + 45*59 + 34*59^2 + 37*59^3 + 29*59^4 + 20*59^5 + 19*59^6 + 2*59^7 + 49*59^8 + 57*59^9+O(59^10) $r_{ 7 }$ $=$ $$18 + 31\cdot 59 + 32\cdot 59^{2} + 31\cdot 59^{3} + 11\cdot 59^{4} + 38\cdot 59^{5} + 7\cdot 59^{6} + 31\cdot 59^{7} + 31\cdot 59^{8} + 38\cdot 59^{9} +O(59^{10})$$ 18 + 31*59 + 32*59^2 + 31*59^3 + 11*59^4 + 38*59^5 + 7*59^6 + 31*59^7 + 31*59^8 + 38*59^9+O(59^10) $r_{ 8 }$ $=$ $$54 a + 18 + \left(12 a + 42\right)\cdot 59 + \left(25 a + 47\right)\cdot 59^{2} + \left(33 a + 24\right)\cdot 59^{3} + \left(21 a + 6\right)\cdot 59^{4} + \left(33 a + 33\right)\cdot 59^{5} + \left(4 a + 10\right)\cdot 59^{6} + \left(13 a + 5\right)\cdot 59^{7} + \left(50 a + 13\right)\cdot 59^{8} + \left(22 a + 45\right)\cdot 59^{9} +O(59^{10})$$ 54*a + 18 + (12*a + 42)*59 + (25*a + 47)*59^2 + (33*a + 24)*59^3 + (21*a + 6)*59^4 + (33*a + 33)*59^5 + (4*a + 10)*59^6 + (13*a + 5)*59^7 + (50*a + 13)*59^8 + (22*a + 45)*59^9+O(59^10)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.