# Properties

 Label 18.508...784.36t1758.a.a Dimension $18$ Group $S_4\wr C_2$ Conductor $5.085\times 10^{30}$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $18$ Group: $S_4\wr C_2$ Conductor: $$508\!\cdots\!784$$$$\medspace = 2^{36} \cdot 3^{22} \cdot 11^{9}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 8.4.32788343808.3 Galois orbit size: $1$ Smallest permutation container: 36T1758 Parity: even Determinant: 1.44.2t1.a.a Projective image: $S_4\wr C_2$ Projective stem field: Galois closure of 8.4.32788343808.3

## Defining polynomial

 $f(x)$ $=$ $$x^{8} + 3x^{6} - 14x^{5} - 18x^{4} + 12x^{3} + 31x^{2} + 18x + 3$$ x^8 + 3*x^6 - 14*x^5 - 18*x^4 + 12*x^3 + 31*x^2 + 18*x + 3 .

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $$x^{2} + 24x + 2$$

Roots:
 $r_{ 1 }$ $=$ $$2 + 20\cdot 29 + 8\cdot 29^{2} + 14\cdot 29^{3} + 13\cdot 29^{4} + 21\cdot 29^{5} + 4\cdot 29^{6} + 5\cdot 29^{7} + 5\cdot 29^{8} + 19\cdot 29^{9} +O(29^{10})$$ 2 + 20*29 + 8*29^2 + 14*29^3 + 13*29^4 + 21*29^5 + 4*29^6 + 5*29^7 + 5*29^8 + 19*29^9+O(29^10) $r_{ 2 }$ $=$ $$15 a + 24 + \left(16 a + 13\right)\cdot 29 + 5\cdot 29^{2} + \left(22 a + 1\right)\cdot 29^{3} + \left(26 a + 20\right)\cdot 29^{4} + \left(20 a + 3\right)\cdot 29^{5} + \left(22 a + 6\right)\cdot 29^{6} + \left(16 a + 25\right)\cdot 29^{7} + \left(5 a + 6\right)\cdot 29^{8} + \left(28 a + 22\right)\cdot 29^{9} +O(29^{10})$$ 15*a + 24 + (16*a + 13)*29 + 5*29^2 + (22*a + 1)*29^3 + (26*a + 20)*29^4 + (20*a + 3)*29^5 + (22*a + 6)*29^6 + (16*a + 25)*29^7 + (5*a + 6)*29^8 + (28*a + 22)*29^9+O(29^10) $r_{ 3 }$ $=$ $$8 + 11\cdot 29 + 20\cdot 29^{2} + 17\cdot 29^{3} + 18\cdot 29^{4} + 6\cdot 29^{5} + 3\cdot 29^{6} + 23\cdot 29^{7} + 23\cdot 29^{8} + 3\cdot 29^{9} +O(29^{10})$$ 8 + 11*29 + 20*29^2 + 17*29^3 + 18*29^4 + 6*29^5 + 3*29^6 + 23*29^7 + 23*29^8 + 3*29^9+O(29^10) $r_{ 4 }$ $=$ $$8 a + 4 + \left(13 a + 13\right)\cdot 29 + \left(10 a + 9\right)\cdot 29^{2} + \left(19 a + 13\right)\cdot 29^{3} + \left(15 a + 12\right)\cdot 29^{4} + \left(21 a + 12\right)\cdot 29^{5} + \left(14 a + 13\right)\cdot 29^{6} + 6\cdot 29^{7} + \left(24 a + 27\right)\cdot 29^{8} + \left(27 a + 17\right)\cdot 29^{9} +O(29^{10})$$ 8*a + 4 + (13*a + 13)*29 + (10*a + 9)*29^2 + (19*a + 13)*29^3 + (15*a + 12)*29^4 + (21*a + 12)*29^5 + (14*a + 13)*29^6 + 6*29^7 + (24*a + 27)*29^8 + (27*a + 17)*29^9+O(29^10) $r_{ 5 }$ $=$ $$21 a + 15 + \left(15 a + 13\right)\cdot 29 + \left(18 a + 19\right)\cdot 29^{2} + \left(9 a + 12\right)\cdot 29^{3} + \left(13 a + 13\right)\cdot 29^{4} + \left(7 a + 17\right)\cdot 29^{5} + \left(14 a + 7\right)\cdot 29^{6} + \left(28 a + 23\right)\cdot 29^{7} + \left(4 a + 1\right)\cdot 29^{8} + \left(a + 17\right)\cdot 29^{9} +O(29^{10})$$ 21*a + 15 + (15*a + 13)*29 + (18*a + 19)*29^2 + (9*a + 12)*29^3 + (13*a + 13)*29^4 + (7*a + 17)*29^5 + (14*a + 7)*29^6 + (28*a + 23)*29^7 + (4*a + 1)*29^8 + (a + 17)*29^9+O(29^10) $r_{ 6 }$ $=$ $$5 a + 13 + \left(4 a + 2\right)\cdot 29 + \left(8 a + 12\right)\cdot 29^{2} + 2 a\cdot 29^{3} + \left(10 a + 16\right)\cdot 29^{4} + \left(8 a + 28\right)\cdot 29^{5} + \left(18 a + 21\right)\cdot 29^{6} + \left(14 a + 3\right)\cdot 29^{7} + \left(21 a + 28\right)\cdot 29^{8} + \left(10 a + 9\right)\cdot 29^{9} +O(29^{10})$$ 5*a + 13 + (4*a + 2)*29 + (8*a + 12)*29^2 + 2*a*29^3 + (10*a + 16)*29^4 + (8*a + 28)*29^5 + (18*a + 21)*29^6 + (14*a + 3)*29^7 + (21*a + 28)*29^8 + (10*a + 9)*29^9+O(29^10) $r_{ 7 }$ $=$ $$24 a + 9 + \left(24 a + 18\right)\cdot 29 + \left(20 a + 19\right)\cdot 29^{2} + \left(26 a + 3\right)\cdot 29^{3} + \left(18 a + 6\right)\cdot 29^{4} + \left(20 a + 2\right)\cdot 29^{5} + \left(10 a + 18\right)\cdot 29^{6} + 14 a\cdot 29^{7} + \left(7 a + 5\right)\cdot 29^{8} + \left(18 a + 13\right)\cdot 29^{9} +O(29^{10})$$ 24*a + 9 + (24*a + 18)*29 + (20*a + 19)*29^2 + (26*a + 3)*29^3 + (18*a + 6)*29^4 + (20*a + 2)*29^5 + (10*a + 18)*29^6 + 14*a*29^7 + (7*a + 5)*29^8 + (18*a + 13)*29^9+O(29^10) $r_{ 8 }$ $=$ $$14 a + 12 + \left(12 a + 23\right)\cdot 29 + \left(28 a + 20\right)\cdot 29^{2} + \left(6 a + 23\right)\cdot 29^{3} + \left(2 a + 15\right)\cdot 29^{4} + \left(8 a + 23\right)\cdot 29^{5} + \left(6 a + 11\right)\cdot 29^{6} + \left(12 a + 28\right)\cdot 29^{7} + \left(23 a + 17\right)\cdot 29^{8} + 12\cdot 29^{9} +O(29^{10})$$ 14*a + 12 + (12*a + 23)*29 + (28*a + 20)*29^2 + (6*a + 23)*29^3 + (2*a + 15)*29^4 + (8*a + 23)*29^5 + (6*a + 11)*29^6 + (12*a + 28)*29^7 + (23*a + 17)*29^8 + 12*29^9+O(29^10)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.