# Properties

 Label 6.340139712.9t31.a.a Dimension $6$ Group $S_3\wr S_3$ Conductor $340139712$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $6$ Group: $S_3\wr S_3$ Conductor: $$340139712$$$$\medspace = 2^{6} \cdot 3 \cdot 11^{6}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 9.1.164627620608.1 Galois orbit size: $1$ Smallest permutation container: $S_3\wr S_3$ Parity: odd Determinant: 1.3.2t1.a.a Projective image: $S_3\wr S_3$ Projective stem field: Galois closure of 9.1.164627620608.1

## Defining polynomial

 $f(x)$ $=$ $$x^{9} - x^{8} - 2x^{7} + 5x^{6} - 10x^{5} + 6x^{4} + 24x^{3} - 20x^{2} - 8x + 24$$ x^9 - x^8 - 2*x^7 + 5*x^6 - 10*x^5 + 6*x^4 + 24*x^3 - 20*x^2 - 8*x + 24 .

The roots of $f$ are computed in an extension of $\Q_{ 97 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$: $$x^{3} + 9x + 92$$

Roots:
 $r_{ 1 }$ $=$ $$91 a^{2} + 87 a + 56 + \left(87 a^{2} + 55 a + 75\right)\cdot 97 + \left(45 a^{2} + 53 a + 64\right)\cdot 97^{2} + \left(17 a^{2} + 61 a + 28\right)\cdot 97^{3} + \left(82 a^{2} + 16 a + 90\right)\cdot 97^{4} + \left(76 a^{2} + 9 a + 4\right)\cdot 97^{5} + \left(55 a^{2} + 43 a + 58\right)\cdot 97^{6} + \left(12 a^{2} + 65 a + 88\right)\cdot 97^{7} + \left(39 a^{2} + 84 a + 72\right)\cdot 97^{8} + \left(95 a^{2} + 4 a + 25\right)\cdot 97^{9} +O(97^{10})$$ 91*a^2 + 87*a + 56 + (87*a^2 + 55*a + 75)*97 + (45*a^2 + 53*a + 64)*97^2 + (17*a^2 + 61*a + 28)*97^3 + (82*a^2 + 16*a + 90)*97^4 + (76*a^2 + 9*a + 4)*97^5 + (55*a^2 + 43*a + 58)*97^6 + (12*a^2 + 65*a + 88)*97^7 + (39*a^2 + 84*a + 72)*97^8 + (95*a^2 + 4*a + 25)*97^9+O(97^10) $r_{ 2 }$ $=$ $$44 a^{2} + 87 a + 72 + \left(56 a^{2} + 94 a + 37\right)\cdot 97 + \left(39 a^{2} + 89 a + 27\right)\cdot 97^{2} + \left(87 a^{2} + 39 a + 9\right)\cdot 97^{3} + \left(89 a^{2} + 40 a + 49\right)\cdot 97^{4} + \left(35 a^{2} + 79 a + 56\right)\cdot 97^{5} + \left(32 a^{2} + 42 a + 61\right)\cdot 97^{6} + \left(56 a^{2} + 76 a + 20\right)\cdot 97^{7} + \left(65 a^{2} + 83 a + 67\right)\cdot 97^{8} + \left(86 a^{2} + 95 a + 68\right)\cdot 97^{9} +O(97^{10})$$ 44*a^2 + 87*a + 72 + (56*a^2 + 94*a + 37)*97 + (39*a^2 + 89*a + 27)*97^2 + (87*a^2 + 39*a + 9)*97^3 + (89*a^2 + 40*a + 49)*97^4 + (35*a^2 + 79*a + 56)*97^5 + (32*a^2 + 42*a + 61)*97^6 + (56*a^2 + 76*a + 20)*97^7 + (65*a^2 + 83*a + 67)*97^8 + (86*a^2 + 95*a + 68)*97^9+O(97^10) $r_{ 3 }$ $=$ $$18 a^{2} + 93 a + 13 + \left(94 a^{2} + 51 a + 70\right)\cdot 97 + \left(78 a^{2} + 90 a + 69\right)\cdot 97^{2} + \left(37 a^{2} + 76 a + 2\right)\cdot 97^{3} + \left(29 a^{2} + 59 a + 74\right)\cdot 97^{4} + \left(82 a^{2} + 65 a + 43\right)\cdot 97^{5} + \left(78 a^{2} + 83 a + 49\right)\cdot 97^{6} + \left(86 a^{2} + 22 a + 9\right)\cdot 97^{7} + \left(34 a^{2} + 7 a + 77\right)\cdot 97^{8} + \left(81 a^{2} + 63 a + 36\right)\cdot 97^{9} +O(97^{10})$$ 18*a^2 + 93*a + 13 + (94*a^2 + 51*a + 70)*97 + (78*a^2 + 90*a + 69)*97^2 + (37*a^2 + 76*a + 2)*97^3 + (29*a^2 + 59*a + 74)*97^4 + (82*a^2 + 65*a + 43)*97^5 + (78*a^2 + 83*a + 49)*97^6 + (86*a^2 + 22*a + 9)*97^7 + (34*a^2 + 7*a + 77)*97^8 + (81*a^2 + 63*a + 36)*97^9+O(97^10) $r_{ 4 }$ $=$ $$69 a^{2} + 43 a + 94 + \left(13 a^{2} + 17 a + 26\right)\cdot 97 + \left(22 a^{2} + 85 a + 36\right)\cdot 97^{2} + \left(45 a^{2} + 48 a + 54\right)\cdot 97^{3} + \left(81 a^{2} + 70 a + 88\right)\cdot 97^{4} + \left(12 a^{2} + 32 a + 77\right)\cdot 97^{5} + \left(39 a^{2} + 22 a + 29\right)\cdot 97^{6} + \left(86 a^{2} + 77 a + 14\right)\cdot 97^{7} + \left(44 a^{2} + 84 a + 46\right)\cdot 97^{8} + \left(5 a^{2} + 81 a + 28\right)\cdot 97^{9} +O(97^{10})$$ 69*a^2 + 43*a + 94 + (13*a^2 + 17*a + 26)*97 + (22*a^2 + 85*a + 36)*97^2 + (45*a^2 + 48*a + 54)*97^3 + (81*a^2 + 70*a + 88)*97^4 + (12*a^2 + 32*a + 77)*97^5 + (39*a^2 + 22*a + 29)*97^6 + (86*a^2 + 77*a + 14)*97^7 + (44*a^2 + 84*a + 46)*97^8 + (5*a^2 + 81*a + 28)*97^9+O(97^10) $r_{ 5 }$ $=$ $$35 a^{2} + 83 a + 11 + \left(49 a^{2} + 79 a + 38\right)\cdot 97 + \left(47 a^{2} + 72 a + 74\right)\cdot 97^{2} + \left(18 a^{2} + 23 a + 34\right)\cdot 97^{3} + \left(27 a^{2} + 64 a + 51\right)\cdot 97^{4} + \left(11 a^{2} + 8 a + 96\right)\cdot 97^{5} + \left(55 a^{2} + 60 a + 53\right)\cdot 97^{6} + \left(78 a^{2} + 53 a + 96\right)\cdot 97^{7} + \left(48 a^{2} + 77 a + 33\right)\cdot 97^{8} + \left(25 a^{2} + 77 a + 91\right)\cdot 97^{9} +O(97^{10})$$ 35*a^2 + 83*a + 11 + (49*a^2 + 79*a + 38)*97 + (47*a^2 + 72*a + 74)*97^2 + (18*a^2 + 23*a + 34)*97^3 + (27*a^2 + 64*a + 51)*97^4 + (11*a^2 + 8*a + 96)*97^5 + (55*a^2 + 60*a + 53)*97^6 + (78*a^2 + 53*a + 96)*97^7 + (48*a^2 + 77*a + 33)*97^8 + (25*a^2 + 77*a + 91)*97^9+O(97^10) $r_{ 6 }$ $=$ $$68 a^{2} + 96 a + 88 + \left(7 a^{2} + 35 a + 87\right)\cdot 97 + \left(72 a^{2} + 52 a + 44\right)\cdot 97^{2} + \left(36 a^{2} + 47 a + 3\right)\cdot 97^{3} + \left(23 a^{2} + 9 a + 31\right)\cdot 97^{4} + \left(69 a^{2} + 30 a + 28\right)\cdot 97^{5} + \left(26 a^{2} + 68 a + 52\right)\cdot 97^{6} + \left(34 a^{2} + 77 a + 89\right)\cdot 97^{7} + \left(45 a^{2} + 70 a + 48\right)\cdot 97^{8} + \left(44 a^{2} + 82 a + 68\right)\cdot 97^{9} +O(97^{10})$$ 68*a^2 + 96*a + 88 + (7*a^2 + 35*a + 87)*97 + (72*a^2 + 52*a + 44)*97^2 + (36*a^2 + 47*a + 3)*97^3 + (23*a^2 + 9*a + 31)*97^4 + (69*a^2 + 30*a + 28)*97^5 + (26*a^2 + 68*a + 52)*97^6 + (34*a^2 + 77*a + 89)*97^7 + (45*a^2 + 70*a + 48)*97^8 + (44*a^2 + 82*a + 68)*97^9+O(97^10) $r_{ 7 }$ $=$ $$35 a^{2} + 14 a + 18 + \left(43 a^{2} + 47 a + 56\right)\cdot 97 + \left(75 a^{2} + 13 a + 48\right)\cdot 97^{2} + \left(68 a^{2} + 77 a + 91\right)\cdot 97^{3} + \left(74 a^{2} + 93 a + 54\right)\cdot 97^{4} + \left(75 a^{2} + 48 a + 4\right)\cdot 97^{5} + \left(82 a^{2} + 67 a + 73\right)\cdot 97^{6} + \left(50 a^{2} + 94 a + 84\right)\cdot 97^{7} + \left(93 a^{2} + 5 a + 40\right)\cdot 97^{8} + \left(25 a^{2} + 35 a + 92\right)\cdot 97^{9} +O(97^{10})$$ 35*a^2 + 14*a + 18 + (43*a^2 + 47*a + 56)*97 + (75*a^2 + 13*a + 48)*97^2 + (68*a^2 + 77*a + 91)*97^3 + (74*a^2 + 93*a + 54)*97^4 + (75*a^2 + 48*a + 4)*97^5 + (82*a^2 + 67*a + 73)*97^6 + (50*a^2 + 94*a + 84)*97^7 + (93*a^2 + 5*a + 40)*97^8 + (25*a^2 + 35*a + 92)*97^9+O(97^10) $r_{ 8 }$ $=$ $$68 a^{2} + 24 a + 15 + \left(56 a^{2} + 58 a + 82\right)\cdot 97 + \left(3 a^{2} + 67 a + 4\right)\cdot 97^{2} + \left(61 a^{2} + 11 a + 96\right)\cdot 97^{3} + \left(84 a^{2} + 16 a + 7\right)\cdot 97^{4} + \left(8 a^{2} + 79 a + 82\right)\cdot 97^{5} + \left(83 a^{2} + 90 a + 27\right)\cdot 97^{6} + \left(5 a^{2} + 74 a + 48\right)\cdot 97^{7} + \left(9 a^{2} + 31 a + 86\right)\cdot 97^{8} + \left(73 a^{2} + 14 a + 85\right)\cdot 97^{9} +O(97^{10})$$ 68*a^2 + 24*a + 15 + (56*a^2 + 58*a + 82)*97 + (3*a^2 + 67*a + 4)*97^2 + (61*a^2 + 11*a + 96)*97^3 + (84*a^2 + 16*a + 7)*97^4 + (8*a^2 + 79*a + 82)*97^5 + (83*a^2 + 90*a + 27)*97^6 + (5*a^2 + 74*a + 48)*97^7 + (9*a^2 + 31*a + 86)*97^8 + (73*a^2 + 14*a + 85)*97^9+O(97^10) $r_{ 9 }$ $=$ $$57 a^{2} + 55 a + 22 + \left(75 a^{2} + 43 a + 10\right)\cdot 97 + \left(2 a^{2} + 56 a + 17\right)\cdot 97^{2} + \left(15 a^{2} + 67\right)\cdot 97^{3} + \left(89 a^{2} + 17 a + 37\right)\cdot 97^{4} + \left(14 a^{2} + 34 a + 90\right)\cdot 97^{5} + \left(31 a^{2} + 6 a + 78\right)\cdot 97^{6} + \left(73 a^{2} + 39 a + 32\right)\cdot 97^{7} + \left(6 a^{2} + 38 a + 11\right)\cdot 97^{8} + \left(47 a^{2} + 29 a + 84\right)\cdot 97^{9} +O(97^{10})$$ 57*a^2 + 55*a + 22 + (75*a^2 + 43*a + 10)*97 + (2*a^2 + 56*a + 17)*97^2 + (15*a^2 + 67)*97^3 + (89*a^2 + 17*a + 37)*97^4 + (14*a^2 + 34*a + 90)*97^5 + (31*a^2 + 6*a + 78)*97^6 + (73*a^2 + 39*a + 32)*97^7 + (6*a^2 + 38*a + 11)*97^8 + (47*a^2 + 29*a + 84)*97^9+O(97^10)

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

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 9 }$ Character value $1$ $1$ $()$ $6$ $9$ $2$ $(1,2)$ $4$ $18$ $2$ $(1,3)(2,4)(5,9)$ $2$ $27$ $2$ $(1,2)(3,4)(6,7)$ $0$ $27$ $2$ $(1,2)(3,4)$ $2$ $54$ $2$ $(1,2)(3,6)(4,7)(5,8)$ $0$ $6$ $3$ $(6,7,8)$ $3$ $8$ $3$ $(1,2,9)(3,4,5)(6,7,8)$ $-3$ $12$ $3$ $(3,4,5)(6,7,8)$ $0$ $72$ $3$ $(1,3,6)(2,4,7)(5,8,9)$ $0$ $54$ $4$ $(1,4,2,3)(5,9)$ $2$ $162$ $4$ $(1,7,2,6)(4,5)(8,9)$ $0$ $36$ $6$ $(1,3)(2,4)(5,9)(6,7,8)$ $-1$ $36$ $6$ $(1,6,2,7,9,8)$ $2$ $36$ $6$ $(1,2)(6,7,8)$ $1$ $36$ $6$ $(1,2)(3,4,5)(6,7,8)$ $-2$ $54$ $6$ $(1,2)(3,4)(6,8,7)$ $-1$ $72$ $6$ $(1,3,2,4,9,5)(6,7,8)$ $-1$ $108$ $6$ $(1,2)(3,6,4,7,5,8)$ $0$ $216$ $6$ $(1,4,7,2,3,6)(5,8,9)$ $0$ $144$ $9$ $(1,3,6,2,4,7,9,5,8)$ $0$ $108$ $12$ $(1,4,2,3)(5,9)(6,7,8)$ $-1$

The blue line marks the conjugacy class containing complex conjugation.