# Properties

 Label 12.332...856.18t315.a.a Dimension $12$ Group $S_3\wr S_3$ Conductor $3.326\times 10^{17}$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $12$ Group: $S_3\wr S_3$ Conductor: $$332552906075897856$$$$\medspace = 2^{15} \cdot 3^{15} \cdot 29^{4}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 9.1.63126687744.1 Galois orbit size: $1$ Smallest permutation container: 18T315 Parity: odd Determinant: 1.24.2t1.b.a Projective image: $C_3^3.S_4.C_2$ Projective stem field: Galois closure of 9.1.63126687744.1

## Defining polynomial

 $f(x)$ $=$ $$x^{9} - 6x^{6} - 9x^{5} - 18x^{4} - 15x^{3} - 18x^{2} - 9x - 8$$ x^9 - 6*x^6 - 9*x^5 - 18*x^4 - 15*x^3 - 18*x^2 - 9*x - 8 .

The roots of $f$ are computed in an extension of $\Q_{ 227 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 227 }$: $$x^{3} + 2x + 225$$

Roots:
 $r_{ 1 }$ $=$ $$112 + 164\cdot 227 + 164\cdot 227^{2} + 60\cdot 227^{3} + 83\cdot 227^{4} + 79\cdot 227^{5} + 46\cdot 227^{6} + 56\cdot 227^{7} + 171\cdot 227^{8} + 57\cdot 227^{9} +O(227^{10})$$ 112 + 164*227 + 164*227^2 + 60*227^3 + 83*227^4 + 79*227^5 + 46*227^6 + 56*227^7 + 171*227^8 + 57*227^9+O(227^10) $r_{ 2 }$ $=$ $$120 + 88\cdot 227 + 191\cdot 227^{2} + 180\cdot 227^{3} + 86\cdot 227^{4} + 12\cdot 227^{5} + 163\cdot 227^{6} + 153\cdot 227^{7} + 120\cdot 227^{8} + 182\cdot 227^{9} +O(227^{10})$$ 120 + 88*227 + 191*227^2 + 180*227^3 + 86*227^4 + 12*227^5 + 163*227^6 + 153*227^7 + 120*227^8 + 182*227^9+O(227^10) $r_{ 3 }$ $=$ $$163 + 202\cdot 227 + 14\cdot 227^{2} + 221\cdot 227^{3} + 37\cdot 227^{4} + 147\cdot 227^{5} + 46\cdot 227^{6} + 57\cdot 227^{7} + 23\cdot 227^{8} + 6\cdot 227^{9} +O(227^{10})$$ 163 + 202*227 + 14*227^2 + 221*227^3 + 37*227^4 + 147*227^5 + 46*227^6 + 57*227^7 + 23*227^8 + 6*227^9+O(227^10) $r_{ 4 }$ $=$ $$16 a^{2} + 77 a + 138 + \left(184 a^{2} + 180 a + 132\right)\cdot 227 + \left(76 a^{2} + 212 a + 28\right)\cdot 227^{2} + \left(85 a^{2} + 8 a + 23\right)\cdot 227^{3} + \left(124 a^{2} + 32 a + 216\right)\cdot 227^{4} + \left(38 a^{2} + 113 a + 87\right)\cdot 227^{5} + \left(14 a^{2} + 90 a + 111\right)\cdot 227^{6} + \left(225 a^{2} + 111 a\right)\cdot 227^{7} + \left(200 a^{2} + 145 a + 74\right)\cdot 227^{8} + \left(107 a^{2} + 81 a + 211\right)\cdot 227^{9} +O(227^{10})$$ 16*a^2 + 77*a + 138 + (184*a^2 + 180*a + 132)*227 + (76*a^2 + 212*a + 28)*227^2 + (85*a^2 + 8*a + 23)*227^3 + (124*a^2 + 32*a + 216)*227^4 + (38*a^2 + 113*a + 87)*227^5 + (14*a^2 + 90*a + 111)*227^6 + (225*a^2 + 111*a)*227^7 + (200*a^2 + 145*a + 74)*227^8 + (107*a^2 + 81*a + 211)*227^9+O(227^10) $r_{ 5 }$ $=$ $$58 a^{2} + 185 a + 194 + \left(171 a^{2} + 60 a + 39\right)\cdot 227 + \left(77 a^{2} + 107 a + 181\right)\cdot 227^{2} + \left(193 a^{2} + 189 a + 15\right)\cdot 227^{3} + \left(175 a^{2} + 221 a + 209\right)\cdot 227^{4} + \left(178 a^{2} + 13 a + 47\right)\cdot 227^{5} + \left(129 a^{2} + 133 a + 114\right)\cdot 227^{6} + \left(118 a^{2} + 213 a + 85\right)\cdot 227^{7} + \left(150 a^{2} + 119 a + 82\right)\cdot 227^{8} + \left(58 a^{2} + 14 a + 221\right)\cdot 227^{9} +O(227^{10})$$ 58*a^2 + 185*a + 194 + (171*a^2 + 60*a + 39)*227 + (77*a^2 + 107*a + 181)*227^2 + (193*a^2 + 189*a + 15)*227^3 + (175*a^2 + 221*a + 209)*227^4 + (178*a^2 + 13*a + 47)*227^5 + (129*a^2 + 133*a + 114)*227^6 + (118*a^2 + 213*a + 85)*227^7 + (150*a^2 + 119*a + 82)*227^8 + (58*a^2 + 14*a + 221)*227^9+O(227^10) $r_{ 6 }$ $=$ $$65 a^{2} + 216 a + 141 + \left(45 a^{2} + 107 a + 172\right)\cdot 227 + \left(206 a^{2} + 171 a + 73\right)\cdot 227^{2} + \left(15 a^{2} + 204 a + 33\right)\cdot 227^{3} + \left(190 a^{2} + 155 a + 58\right)\cdot 227^{4} + \left(207 a^{2} + 26 a + 85\right)\cdot 227^{5} + \left(140 a^{2} + 174 a + 161\right)\cdot 227^{6} + \left(72 a^{2} + 6 a + 4\right)\cdot 227^{7} + \left(10 a^{2} + 123 a + 27\right)\cdot 227^{8} + \left(168 a^{2} + 61 a + 150\right)\cdot 227^{9} +O(227^{10})$$ 65*a^2 + 216*a + 141 + (45*a^2 + 107*a + 172)*227 + (206*a^2 + 171*a + 73)*227^2 + (15*a^2 + 204*a + 33)*227^3 + (190*a^2 + 155*a + 58)*227^4 + (207*a^2 + 26*a + 85)*227^5 + (140*a^2 + 174*a + 161)*227^6 + (72*a^2 + 6*a + 4)*227^7 + (10*a^2 + 123*a + 27)*227^8 + (168*a^2 + 61*a + 150)*227^9+O(227^10) $r_{ 7 }$ $=$ $$153 a^{2} + 192 a + 18 + \left(98 a^{2} + 212 a + 170\right)\cdot 227 + \left(72 a^{2} + 133 a + 22\right)\cdot 227^{2} + \left(175 a^{2} + 28 a + 143\right)\cdot 227^{3} + \left(153 a^{2} + 200 a + 179\right)\cdot 227^{4} + \left(9 a^{2} + 99 a + 200\right)\cdot 227^{5} + \left(83 a^{2} + 3 a + 51\right)\cdot 227^{6} + \left(110 a^{2} + 129 a + 150\right)\cdot 227^{7} + \left(102 a^{2} + 188 a + 169\right)\cdot 227^{8} + \left(60 a^{2} + 130 a + 223\right)\cdot 227^{9} +O(227^{10})$$ 153*a^2 + 192*a + 18 + (98*a^2 + 212*a + 170)*227 + (72*a^2 + 133*a + 22)*227^2 + (175*a^2 + 28*a + 143)*227^3 + (153*a^2 + 200*a + 179)*227^4 + (9*a^2 + 99*a + 200)*227^5 + (83*a^2 + 3*a + 51)*227^6 + (110*a^2 + 129*a + 150)*227^7 + (102*a^2 + 188*a + 169)*227^8 + (60*a^2 + 130*a + 223)*227^9+O(227^10) $r_{ 8 }$ $=$ $$169 a^{2} + 107 a + 204 + \left(10 a + 188\right)\cdot 227 + \left(157 a^{2} + 186 a + 83\right)\cdot 227^{2} + \left(45 a^{2} + 114 a + 224\right)\cdot 227^{3} + \left(52 a^{2} + 98 a + 176\right)\cdot 227^{4} + \left(40 a^{2} + 5 a + 88\right)\cdot 227^{5} + \left(158 a^{2} + 116 a + 184\right)\cdot 227^{6} + \left(98 a^{2} + 120 a + 190\right)\cdot 227^{7} + \left(125 a^{2} + 11 a + 104\right)\cdot 227^{8} + \left(172 a^{2} + 55 a + 80\right)\cdot 227^{9} +O(227^{10})$$ 169*a^2 + 107*a + 204 + (10*a + 188)*227 + (157*a^2 + 186*a + 83)*227^2 + (45*a^2 + 114*a + 224)*227^3 + (52*a^2 + 98*a + 176)*227^4 + (40*a^2 + 5*a + 88)*227^5 + (158*a^2 + 116*a + 184)*227^6 + (98*a^2 + 120*a + 190)*227^7 + (125*a^2 + 11*a + 104)*227^8 + (172*a^2 + 55*a + 80)*227^9+O(227^10) $r_{ 9 }$ $=$ $$220 a^{2} + 131 a + 45 + \left(180 a^{2} + 108 a + 202\right)\cdot 227 + \left(90 a^{2} + 96 a + 146\right)\cdot 227^{2} + \left(165 a^{2} + 134 a + 5\right)\cdot 227^{3} + \left(211 a^{2} + 199 a + 87\right)\cdot 227^{4} + \left(205 a^{2} + 194 a + 158\right)\cdot 227^{5} + \left(154 a^{2} + 163 a + 28\right)\cdot 227^{6} + \left(55 a^{2} + 99 a + 209\right)\cdot 227^{7} + \left(91 a^{2} + 92 a + 134\right)\cdot 227^{8} + \left(113 a^{2} + 110 a + 1\right)\cdot 227^{9} +O(227^{10})$$ 220*a^2 + 131*a + 45 + (180*a^2 + 108*a + 202)*227 + (90*a^2 + 96*a + 146)*227^2 + (165*a^2 + 134*a + 5)*227^3 + (211*a^2 + 199*a + 87)*227^4 + (205*a^2 + 194*a + 158)*227^5 + (154*a^2 + 163*a + 28)*227^6 + (55*a^2 + 99*a + 209)*227^7 + (91*a^2 + 92*a + 134)*227^8 + (113*a^2 + 110*a + 1)*227^9+O(227^10)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.