# Properties

 Label 8.98673100992.9t26.a.a Dimension $8$ Group $((C_3^2:Q_8):C_3):C_2$ Conductor $98673100992$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $8$ Group: $((C_3^2:Q_8):C_3):C_2$ Conductor: $$98673100992$$$$\medspace = 2^{6} \cdot 3^{7} \cdot 89^{3}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 9.3.98673100992.1 Galois orbit size: $1$ Smallest permutation container: $((C_3^2:Q_8):C_3):C_2$ Parity: odd Determinant: 1.267.2t1.a.a Projective image: $C_3^2:\GL(2,3)$ Projective stem field: Galois closure of 9.3.98673100992.1

## Defining polynomial

 $f(x)$ $=$ $$x^{9} - 3x^{8} + 3x^{7} - 6x^{6} + 12x^{5} + 3x^{4} - 21x^{3} + 9x^{2} + 1$$ x^9 - 3*x^8 + 3*x^7 - 6*x^6 + 12*x^5 + 3*x^4 - 21*x^3 + 9*x^2 + 1 .

The roots of $f$ are computed in an extension of $\Q_{ 101 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 101 }$: $$x^{4} + x^{2} + 78x + 2$$

Roots:
 $r_{ 1 }$ $=$ $$81 + 42\cdot 101 + 48\cdot 101^{2} + 91\cdot 101^{3} + 20\cdot 101^{4} + 63\cdot 101^{5} + 96\cdot 101^{6} + 55\cdot 101^{7} + 95\cdot 101^{8} + 55\cdot 101^{9} +O(101^{10})$$ 81 + 42*101 + 48*101^2 + 91*101^3 + 20*101^4 + 63*101^5 + 96*101^6 + 55*101^7 + 95*101^8 + 55*101^9+O(101^10) $r_{ 2 }$ $=$ $$64 a^{3} + 4 a^{2} + 32 a + 6 + \left(86 a^{3} + a^{2} + 67 a + 40\right)\cdot 101 + \left(78 a^{3} + 36 a^{2} + 7 a + 69\right)\cdot 101^{2} + \left(14 a^{3} + 10 a^{2} + 99 a + 19\right)\cdot 101^{3} + \left(68 a^{3} + 48 a^{2} + 32 a + 84\right)\cdot 101^{4} + \left(7 a^{3} + 39 a^{2} + 31 a + 27\right)\cdot 101^{5} + \left(67 a^{3} + 17 a^{2} + 92 a + 25\right)\cdot 101^{6} + \left(97 a^{3} + 16 a^{2} + 8 a + 65\right)\cdot 101^{7} + \left(a^{3} + 55 a^{2} + 39 a + 77\right)\cdot 101^{8} + \left(33 a^{3} + 79 a^{2} + 73 a + 76\right)\cdot 101^{9} +O(101^{10})$$ 64*a^3 + 4*a^2 + 32*a + 6 + (86*a^3 + a^2 + 67*a + 40)*101 + (78*a^3 + 36*a^2 + 7*a + 69)*101^2 + (14*a^3 + 10*a^2 + 99*a + 19)*101^3 + (68*a^3 + 48*a^2 + 32*a + 84)*101^4 + (7*a^3 + 39*a^2 + 31*a + 27)*101^5 + (67*a^3 + 17*a^2 + 92*a + 25)*101^6 + (97*a^3 + 16*a^2 + 8*a + 65)*101^7 + (a^3 + 55*a^2 + 39*a + 77)*101^8 + (33*a^3 + 79*a^2 + 73*a + 76)*101^9+O(101^10) $r_{ 3 }$ $=$ $$99 a^{3} + 80 a^{2} + 15 a + 58 + \left(81 a^{3} + 99 a^{2} + 62 a + 66\right)\cdot 101 + \left(54 a^{3} + 70 a^{2} + 79 a + 14\right)\cdot 101^{2} + \left(67 a^{3} + 16 a^{2} + 18 a + 65\right)\cdot 101^{3} + \left(49 a^{3} + 69 a^{2} + 4 a + 18\right)\cdot 101^{4} + \left(90 a^{3} + 34 a^{2} + 12 a + 80\right)\cdot 101^{5} + \left(91 a^{3} + 4 a^{2} + 64 a + 19\right)\cdot 101^{6} + \left(82 a^{3} + 54 a^{2} + 87 a + 91\right)\cdot 101^{7} + \left(100 a^{3} + 43 a^{2} + 70 a + 1\right)\cdot 101^{8} + \left(31 a^{3} + 59 a^{2} + 90 a + 20\right)\cdot 101^{9} +O(101^{10})$$ 99*a^3 + 80*a^2 + 15*a + 58 + (81*a^3 + 99*a^2 + 62*a + 66)*101 + (54*a^3 + 70*a^2 + 79*a + 14)*101^2 + (67*a^3 + 16*a^2 + 18*a + 65)*101^3 + (49*a^3 + 69*a^2 + 4*a + 18)*101^4 + (90*a^3 + 34*a^2 + 12*a + 80)*101^5 + (91*a^3 + 4*a^2 + 64*a + 19)*101^6 + (82*a^3 + 54*a^2 + 87*a + 91)*101^7 + (100*a^3 + 43*a^2 + 70*a + 1)*101^8 + (31*a^3 + 59*a^2 + 90*a + 20)*101^9+O(101^10) $r_{ 4 }$ $=$ $$61 a^{3} + 4 a^{2} + 46 a + 83 + \left(4 a^{2} + 67 a + 58\right)\cdot 101 + \left(8 a^{3} + 16 a^{2} + 78 a + 55\right)\cdot 101^{2} + \left(82 a^{3} + 28 a^{2} + 57 a + 26\right)\cdot 101^{3} + \left(20 a^{3} + 72 a^{2} + 41 a + 54\right)\cdot 101^{4} + \left(14 a^{3} + 91 a^{2} + 79 a + 82\right)\cdot 101^{5} + \left(43 a^{3} + 72 a^{2} + 72 a + 66\right)\cdot 101^{6} + \left(80 a^{3} + 78 a^{2} + 19 a + 47\right)\cdot 101^{7} + \left(10 a^{3} + 46 a^{2} + 9 a + 34\right)\cdot 101^{8} + \left(2 a^{3} + 18 a^{2} + 22 a + 5\right)\cdot 101^{9} +O(101^{10})$$ 61*a^3 + 4*a^2 + 46*a + 83 + (4*a^2 + 67*a + 58)*101 + (8*a^3 + 16*a^2 + 78*a + 55)*101^2 + (82*a^3 + 28*a^2 + 57*a + 26)*101^3 + (20*a^3 + 72*a^2 + 41*a + 54)*101^4 + (14*a^3 + 91*a^2 + 79*a + 82)*101^5 + (43*a^3 + 72*a^2 + 72*a + 66)*101^6 + (80*a^3 + 78*a^2 + 19*a + 47)*101^7 + (10*a^3 + 46*a^2 + 9*a + 34)*101^8 + (2*a^3 + 18*a^2 + 22*a + 5)*101^9+O(101^10) $r_{ 5 }$ $=$ $$30 a^{3} + 44 a^{2} + 61 a + 94 + \left(98 a^{3} + 70 a^{2} + 62 a + 46\right)\cdot 101 + \left(21 a^{3} + 47 a^{2} + 66 a + 51\right)\cdot 101^{2} + \left(48 a^{3} + 10 a^{2} + 87 a + 42\right)\cdot 101^{3} + \left(62 a^{3} + 62 a^{2} + 9 a + 57\right)\cdot 101^{4} + \left(96 a^{3} + 27 a^{2} + 41 a + 81\right)\cdot 101^{5} + \left(82 a^{3} + 73 a^{2} + 100 a + 86\right)\cdot 101^{6} + \left(54 a^{3} + 49 a^{2} + 21 a + 61\right)\cdot 101^{7} + \left(38 a^{3} + 24 a^{2} + 24 a + 35\right)\cdot 101^{8} + \left(99 a^{3} + 62 a^{2} + 96 a + 24\right)\cdot 101^{9} +O(101^{10})$$ 30*a^3 + 44*a^2 + 61*a + 94 + (98*a^3 + 70*a^2 + 62*a + 46)*101 + (21*a^3 + 47*a^2 + 66*a + 51)*101^2 + (48*a^3 + 10*a^2 + 87*a + 42)*101^3 + (62*a^3 + 62*a^2 + 9*a + 57)*101^4 + (96*a^3 + 27*a^2 + 41*a + 81)*101^5 + (82*a^3 + 73*a^2 + 100*a + 86)*101^6 + (54*a^3 + 49*a^2 + 21*a + 61)*101^7 + (38*a^3 + 24*a^2 + 24*a + 35)*101^8 + (99*a^3 + 62*a^2 + 96*a + 24)*101^9+O(101^10) $r_{ 6 }$ $=$ $$35 a^{3} + 71 a^{2} + 71 a + 60 + \left(69 a^{3} + 98 a^{2} + 43 a + 87\right)\cdot 101 + \left(65 a^{3} + 65 a^{2} + 51 a + 20\right)\cdot 101^{2} + \left(74 a^{3} + 13 a^{2} + 31 a + 39\right)\cdot 101^{3} + \left(75 a^{3} + 93 a^{2} + 16 a + 96\right)\cdot 101^{4} + \left(67 a^{3} + 26 a^{2} + 89 a\right)\cdot 101^{5} + \left(15 a^{3} + 7 a^{2} + 50 a + 43\right)\cdot 101^{6} + \left(22 a^{3} + 64 a^{2} + 32 a + 65\right)\cdot 101^{7} + \left(86 a^{3} + 42 a^{2} + 82 a\right)\cdot 101^{8} + \left(57 a^{3} + 44 a^{2} + 100 a + 98\right)\cdot 101^{9} +O(101^{10})$$ 35*a^3 + 71*a^2 + 71*a + 60 + (69*a^3 + 98*a^2 + 43*a + 87)*101 + (65*a^3 + 65*a^2 + 51*a + 20)*101^2 + (74*a^3 + 13*a^2 + 31*a + 39)*101^3 + (75*a^3 + 93*a^2 + 16*a + 96)*101^4 + (67*a^3 + 26*a^2 + 89*a)*101^5 + (15*a^3 + 7*a^2 + 50*a + 43)*101^6 + (22*a^3 + 64*a^2 + 32*a + 65)*101^7 + (86*a^3 + 42*a^2 + 82*a)*101^8 + (57*a^3 + 44*a^2 + 100*a + 98)*101^9+O(101^10) $r_{ 7 }$ $=$ $$4 a^{3} + 47 a^{2} + 84 a + 39 + \left(19 a^{3} + 10 a^{2} + 74 a + 26\right)\cdot 101 + \left(25 a^{3} + 14 a^{2} + 20 a + 71\right)\cdot 101^{2} + \left(74 a^{3} + 34 a^{2} + 26 a + 11\right)\cdot 101^{3} + \left(27 a^{3} + 63 a^{2} + 37 a + 96\right)\cdot 101^{4} + \left(23 a^{3} + 79 a^{2} + 65 a + 58\right)\cdot 101^{5} + \left(18 a^{3} + 29 a^{2} + 49 a + 15\right)\cdot 101^{6} + \left(21 a^{3} + 89 a^{2} + 74 a + 83\right)\cdot 101^{7} + \left(46 a^{3} + 9 a^{2} + 10 a + 72\right)\cdot 101^{8} + \left(71 a^{3} + 18 a^{2} + 60 a + 85\right)\cdot 101^{9} +O(101^{10})$$ 4*a^3 + 47*a^2 + 84*a + 39 + (19*a^3 + 10*a^2 + 74*a + 26)*101 + (25*a^3 + 14*a^2 + 20*a + 71)*101^2 + (74*a^3 + 34*a^2 + 26*a + 11)*101^3 + (27*a^3 + 63*a^2 + 37*a + 96)*101^4 + (23*a^3 + 79*a^2 + 65*a + 58)*101^5 + (18*a^3 + 29*a^2 + 49*a + 15)*101^6 + (21*a^3 + 89*a^2 + 74*a + 83)*101^7 + (46*a^3 + 9*a^2 + 10*a + 72)*101^8 + (71*a^3 + 18*a^2 + 60*a + 85)*101^9+O(101^10) $r_{ 8 }$ $=$ $$42 a^{3} + 22 a^{2} + 53 a + 41 + \left(45 a^{3} + 98 a^{2} + 23 a\right)\cdot 101 + \left(49 a^{3} + 83 a^{2} + 64 a + 90\right)\cdot 101^{2} + \left(30 a^{3} + 48 a^{2} + 13 a + 48\right)\cdot 101^{3} + \left(37 a^{3} + 89 a^{2} + 10 a + 17\right)\cdot 101^{4} + \left(11 a^{3} + 43 a^{2} + 2 a + 44\right)\cdot 101^{5} + \left(76 a^{3} + 3 a^{2} + 87 a + 92\right)\cdot 101^{6} + \left(a^{3} + 43 a^{2} + 39 a + 22\right)\cdot 101^{7} + \left(2 a^{3} + 57 a^{2} + 71 a + 31\right)\cdot 101^{8} + \left(8 a^{3} + 59 a^{2} + 5 a + 18\right)\cdot 101^{9} +O(101^{10})$$ 42*a^3 + 22*a^2 + 53*a + 41 + (45*a^3 + 98*a^2 + 23*a)*101 + (49*a^3 + 83*a^2 + 64*a + 90)*101^2 + (30*a^3 + 48*a^2 + 13*a + 48)*101^3 + (37*a^3 + 89*a^2 + 10*a + 17)*101^4 + (11*a^3 + 43*a^2 + 2*a + 44)*101^5 + (76*a^3 + 3*a^2 + 87*a + 92)*101^6 + (a^3 + 43*a^2 + 39*a + 22)*101^7 + (2*a^3 + 57*a^2 + 71*a + 31)*101^8 + (8*a^3 + 59*a^2 + 5*a + 18)*101^9+O(101^10) $r_{ 9 }$ $=$ $$69 a^{3} + 31 a^{2} + 42 a + 46 + \left(2 a^{3} + 21 a^{2} + 2 a + 34\right)\cdot 101 + \left(100 a^{3} + 69 a^{2} + 35 a + 83\right)\cdot 101^{2} + \left(11 a^{3} + 39 a^{2} + 69 a + 58\right)\cdot 101^{3} + \left(62 a^{3} + 7 a^{2} + 49 a + 59\right)\cdot 101^{4} + \left(92 a^{3} + 60 a^{2} + 83 a + 65\right)\cdot 101^{5} + \left(8 a^{3} + 94 a^{2} + 88 a + 58\right)\cdot 101^{6} + \left(43 a^{3} + 8 a^{2} + 17 a + 11\right)\cdot 101^{7} + \left(16 a^{3} + 23 a^{2} + 96 a + 54\right)\cdot 101^{8} + \left(100 a^{3} + 62 a^{2} + 55 a + 19\right)\cdot 101^{9} +O(101^{10})$$ 69*a^3 + 31*a^2 + 42*a + 46 + (2*a^3 + 21*a^2 + 2*a + 34)*101 + (100*a^3 + 69*a^2 + 35*a + 83)*101^2 + (11*a^3 + 39*a^2 + 69*a + 58)*101^3 + (62*a^3 + 7*a^2 + 49*a + 59)*101^4 + (92*a^3 + 60*a^2 + 83*a + 65)*101^5 + (8*a^3 + 94*a^2 + 88*a + 58)*101^6 + (43*a^3 + 8*a^2 + 17*a + 11)*101^7 + (16*a^3 + 23*a^2 + 96*a + 54)*101^8 + (100*a^3 + 62*a^2 + 55*a + 19)*101^9+O(101^10)

## Generators of the action on the roots $r_1, \ldots, r_{ 9 }$

 Cycle notation $(2,8,3)(4,7,6)$ $(2,8,6,4)(3,9,7,5)$ $(1,2,6)(3,4,9)(5,8,7)$ $(3,9,4)(5,8,7)$ $(1,5,9)(2,8,3)(4,6,7)$ $(1,6,3,7,4,9,2,5)$

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 9 }$ Character value $1$ $1$ $()$ $8$ $9$ $2$ $(1,4)(2,3)(5,7)(6,9)$ $0$ $36$ $2$ $(1,4)(5,6)(7,9)$ $2$ $8$ $3$ $(1,2,6)(3,4,9)(5,8,7)$ $-1$ $24$ $3$ $(1,7,3)(2,4,5)$ $2$ $48$ $3$ $(1,7,9)(2,5,3)(4,6,8)$ $-1$ $54$ $4$ $(1,3,4,2)(5,6,7,9)$ $0$ $72$ $6$ $(1,6,9,4,5,7)(2,8,3)$ $-1$ $72$ $6$ $(1,7)(2,6,4,8,5,9)$ $0$ $54$ $8$ $(1,6,3,7,4,9,2,5)$ $0$ $54$ $8$ $(1,9,3,5,4,6,2,7)$ $0$

The blue line marks the conjugacy class containing complex conjugation.