# Properties

 Label 12.428...752.18t315.a.a Dimension $12$ Group $S_3\wr S_3$ Conductor $4.287\times 10^{16}$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $12$ Group: $S_3\wr S_3$ Conductor: $$42865179747426752$$$$\medspace = 2^{6} \cdot 23^{5} \cdot 101^{4}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 9.1.1808892224.1 Galois orbit size: $1$ Smallest permutation container: 18T315 Parity: odd Determinant: 1.23.2t1.a.a Projective image: $S_3\wr S_3$ Projective stem field: Galois closure of 9.1.1808892224.1

## Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 131 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 131 }$: $$x^{3} + 3x + 129$$

Roots:
 $r_{ 1 }$ $=$ $$28 a^{2} + 106 a + 27 + \left(16 a^{2} + 49 a + 87\right)\cdot 131 + \left(12 a^{2} + 110 a + 112\right)\cdot 131^{2} + \left(96 a^{2} + 60 a + 69\right)\cdot 131^{3} + \left(31 a^{2} + 3 a + 49\right)\cdot 131^{4} + \left(62 a^{2} + 60 a + 11\right)\cdot 131^{5} + \left(114 a^{2} + 101 a + 1\right)\cdot 131^{6} + \left(59 a^{2} + 105 a + 121\right)\cdot 131^{7} + \left(16 a^{2} + 78 a + 4\right)\cdot 131^{8} + \left(84 a^{2} + 19 a + 29\right)\cdot 131^{9} +O(131^{10})$$ 28*a^2 + 106*a + 27 + (16*a^2 + 49*a + 87)*131 + (12*a^2 + 110*a + 112)*131^2 + (96*a^2 + 60*a + 69)*131^3 + (31*a^2 + 3*a + 49)*131^4 + (62*a^2 + 60*a + 11)*131^5 + (114*a^2 + 101*a + 1)*131^6 + (59*a^2 + 105*a + 121)*131^7 + (16*a^2 + 78*a + 4)*131^8 + (84*a^2 + 19*a + 29)*131^9+O(131^10) $r_{ 2 }$ $=$ $$31 a^{2} + a + 116 + \left(113 a^{2} + 12 a + 88\right)\cdot 131 + \left(98 a^{2} + 83 a + 67\right)\cdot 131^{2} + \left(15 a^{2} + 38 a + 79\right)\cdot 131^{3} + \left(2 a^{2} + 27 a + 37\right)\cdot 131^{4} + \left(15 a^{2} + 27 a + 63\right)\cdot 131^{5} + \left(90 a^{2} + 55 a + 125\right)\cdot 131^{6} + \left(2 a^{2} + 35 a + 101\right)\cdot 131^{7} + \left(48 a^{2} + 60 a + 33\right)\cdot 131^{8} + \left(111 a^{2} + 96 a + 81\right)\cdot 131^{9} +O(131^{10})$$ 31*a^2 + a + 116 + (113*a^2 + 12*a + 88)*131 + (98*a^2 + 83*a + 67)*131^2 + (15*a^2 + 38*a + 79)*131^3 + (2*a^2 + 27*a + 37)*131^4 + (15*a^2 + 27*a + 63)*131^5 + (90*a^2 + 55*a + 125)*131^6 + (2*a^2 + 35*a + 101)*131^7 + (48*a^2 + 60*a + 33)*131^8 + (111*a^2 + 96*a + 81)*131^9+O(131^10) $r_{ 3 }$ $=$ $$33 a^{2} + 64 a + 41 + \left(84 a^{2} + 4 a + 120\right)\cdot 131 + \left(65 a^{2} + 129 a + 41\right)\cdot 131^{2} + \left(a^{2} + 24 a + 77\right)\cdot 131^{3} + \left(48 a^{2} + 86 a + 76\right)\cdot 131^{4} + \left(92 a^{2} + 115 a + 2\right)\cdot 131^{5} + \left(95 a^{2} + 70 a + 81\right)\cdot 131^{6} + \left(48 a^{2} + 66 a + 130\right)\cdot 131^{7} + \left(a^{2} + 2 a + 92\right)\cdot 131^{8} + \left(45 a^{2} + 35 a + 108\right)\cdot 131^{9} +O(131^{10})$$ 33*a^2 + 64*a + 41 + (84*a^2 + 4*a + 120)*131 + (65*a^2 + 129*a + 41)*131^2 + (a^2 + 24*a + 77)*131^3 + (48*a^2 + 86*a + 76)*131^4 + (92*a^2 + 115*a + 2)*131^5 + (95*a^2 + 70*a + 81)*131^6 + (48*a^2 + 66*a + 130)*131^7 + (a^2 + 2*a + 92)*131^8 + (45*a^2 + 35*a + 108)*131^9+O(131^10) $r_{ 4 }$ $=$ $$36 a^{2} + 90 a + 43 + \left(50 a^{2} + 97 a + 24\right)\cdot 131 + \left(21 a^{2} + 101 a\right)\cdot 131^{2} + \left(52 a^{2} + 2 a + 113\right)\cdot 131^{3} + \left(18 a^{2} + 110 a + 22\right)\cdot 131^{4} + \left(45 a^{2} + 82 a + 108\right)\cdot 131^{5} + \left(71 a^{2} + 24 a + 45\right)\cdot 131^{6} + \left(122 a^{2} + 127 a + 115\right)\cdot 131^{7} + \left(32 a^{2} + 114 a + 37\right)\cdot 131^{8} + \left(72 a^{2} + 111 a + 5\right)\cdot 131^{9} +O(131^{10})$$ 36*a^2 + 90*a + 43 + (50*a^2 + 97*a + 24)*131 + (21*a^2 + 101*a)*131^2 + (52*a^2 + 2*a + 113)*131^3 + (18*a^2 + 110*a + 22)*131^4 + (45*a^2 + 82*a + 108)*131^5 + (71*a^2 + 24*a + 45)*131^6 + (122*a^2 + 127*a + 115)*131^7 + (32*a^2 + 114*a + 37)*131^8 + (72*a^2 + 111*a + 5)*131^9+O(131^10) $r_{ 5 }$ $=$ $$39 a^{2} + 68 a + 1 + \left(74 a^{2} + 118 a + 11\right)\cdot 131 + \left(107 a^{2} + 101 a + 85\right)\cdot 131^{2} + \left(125 a^{2} + 32 a + 37\right)\cdot 131^{3} + \left(84 a^{2} + 108 a + 72\right)\cdot 131^{4} + \left(110 a + 34\right)\cdot 131^{5} + \left(68 a^{2} + 17 a + 81\right)\cdot 131^{6} + \left(34 a^{2} + 20 a + 34\right)\cdot 131^{7} + \left(94 a^{2} + 47 a + 126\right)\cdot 131^{8} + \left(83 a^{2} + 7 a + 25\right)\cdot 131^{9} +O(131^{10})$$ 39*a^2 + 68*a + 1 + (74*a^2 + 118*a + 11)*131 + (107*a^2 + 101*a + 85)*131^2 + (125*a^2 + 32*a + 37)*131^3 + (84*a^2 + 108*a + 72)*131^4 + (110*a + 34)*131^5 + (68*a^2 + 17*a + 81)*131^6 + (34*a^2 + 20*a + 34)*131^7 + (94*a^2 + 47*a + 126)*131^8 + (83*a^2 + 7*a + 25)*131^9+O(131^10) $r_{ 6 }$ $=$ $$42 a^{2} + 94 a + 59 + \left(40 a^{2} + 80 a + 32\right)\cdot 131 + \left(63 a^{2} + 74 a + 37\right)\cdot 131^{2} + \left(45 a^{2} + 10 a + 34\right)\cdot 131^{3} + \left(55 a^{2} + a + 91\right)\cdot 131^{4} + \left(84 a^{2} + 78 a + 117\right)\cdot 131^{5} + \left(43 a^{2} + 102 a + 107\right)\cdot 131^{6} + \left(108 a^{2} + 80 a + 118\right)\cdot 131^{7} + \left(125 a^{2} + 28 a + 79\right)\cdot 131^{8} + \left(110 a^{2} + 84 a + 109\right)\cdot 131^{9} +O(131^{10})$$ 42*a^2 + 94*a + 59 + (40*a^2 + 80*a + 32)*131 + (63*a^2 + 74*a + 37)*131^2 + (45*a^2 + 10*a + 34)*131^3 + (55*a^2 + a + 91)*131^4 + (84*a^2 + 78*a + 117)*131^5 + (43*a^2 + 102*a + 107)*131^6 + (108*a^2 + 80*a + 118)*131^7 + (125*a^2 + 28*a + 79)*131^8 + (110*a^2 + 84*a + 109)*131^9+O(131^10) $r_{ 7 }$ $=$ $$56 a^{2} + 104 a + 87 + \left(6 a^{2} + 45 a + 95\right)\cdot 131 + \left(2 a^{2} + 58 a + 45\right)\cdot 131^{2} + \left(84 a^{2} + 95 a + 111\right)\cdot 131^{3} + \left(27 a^{2} + 43 a + 35\right)\cdot 131^{4} + \left(85 a^{2} + 68 a + 119\right)\cdot 131^{5} + \left(122 a^{2} + 88 a + 3\right)\cdot 131^{6} + \left(104 a^{2} + 114 a + 112\right)\cdot 131^{7} + \left(3 a^{2} + 99 a + 97\right)\cdot 131^{8} + \left(106 a^{2} + 11 a + 99\right)\cdot 131^{9} +O(131^{10})$$ 56*a^2 + 104*a + 87 + (6*a^2 + 45*a + 95)*131 + (2*a^2 + 58*a + 45)*131^2 + (84*a^2 + 95*a + 111)*131^3 + (27*a^2 + 43*a + 35)*131^4 + (85*a^2 + 68*a + 119)*131^5 + (122*a^2 + 88*a + 3)*131^6 + (104*a^2 + 114*a + 112)*131^7 + (3*a^2 + 99*a + 97)*131^8 + (106*a^2 + 11*a + 99)*131^9+O(131^10) $r_{ 8 }$ $=$ $$61 a^{2} + 62 a + 45 + \left(74 a^{2} + 11\right)\cdot 131 + \left(55 a^{2} + 77 a + 112\right)\cdot 131^{2} + \left(120 a^{2} + 59 a + 26\right)\cdot 131^{3} + \left(43 a^{2} + 126 a + 121\right)\cdot 131^{4} + \left(115 a^{2} + 123 a + 1\right)\cdot 131^{5} + \left(103 a^{2} + 57 a + 22\right)\cdot 131^{6} + \left(93 a^{2} + 75 a + 22\right)\cdot 131^{7} + \left(119 a^{2} + 23 a + 46\right)\cdot 131^{8} + \left(66 a^{2} + 27 a + 123\right)\cdot 131^{9} +O(131^{10})$$ 61*a^2 + 62*a + 45 + (74*a^2 + 11)*131 + (55*a^2 + 77*a + 112)*131^2 + (120*a^2 + 59*a + 26)*131^3 + (43*a^2 + 126*a + 121)*131^4 + (115*a^2 + 123*a + 1)*131^5 + (103*a^2 + 57*a + 22)*131^6 + (93*a^2 + 75*a + 22)*131^7 + (119*a^2 + 23*a + 46)*131^8 + (66*a^2 + 27*a + 123)*131^9+O(131^10) $r_{ 9 }$ $=$ $$67 a^{2} + 66 a + 105 + \left(64 a^{2} + 114 a + 52\right)\cdot 131 + \left(97 a^{2} + 49 a + 21\right)\cdot 131^{2} + \left(113 a^{2} + 67 a + 105\right)\cdot 131^{3} + \left(80 a^{2} + 17 a + 16\right)\cdot 131^{4} + \left(23 a^{2} + 119 a + 65\right)\cdot 131^{5} + \left(76 a^{2} + 4 a + 55\right)\cdot 131^{6} + \left(79 a^{2} + 29 a + 29\right)\cdot 131^{7} + \left(81 a^{2} + 68 a + 4\right)\cdot 131^{8} + \left(105 a^{2} + 130 a + 72\right)\cdot 131^{9} +O(131^{10})$$ 67*a^2 + 66*a + 105 + (64*a^2 + 114*a + 52)*131 + (97*a^2 + 49*a + 21)*131^2 + (113*a^2 + 67*a + 105)*131^3 + (80*a^2 + 17*a + 16)*131^4 + (23*a^2 + 119*a + 65)*131^5 + (76*a^2 + 4*a + 55)*131^6 + (79*a^2 + 29*a + 29)*131^7 + (81*a^2 + 68*a + 4)*131^8 + (105*a^2 + 130*a + 72)*131^9+O(131^10)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.