# Properties

 Label 12.335...000.36t1123.a.a Dimension $12$ Group $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ Conductor $3.355\times 10^{17}$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $12$ Group: $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ Conductor: $$335544320000000000$$$$\medspace = 2^{35} \cdot 5^{10}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 9.1.25600000000.1 Galois orbit size: $1$ Smallest permutation container: 36T1123 Parity: odd Determinant: 1.8.2t1.b.a Projective image: $C_3^3:S_4$ Projective stem field: Galois closure of 9.1.25600000000.1

## Defining polynomial

 $f(x)$ $=$ $$x^{9} - 2x^{8} + 4x^{7} - 8x^{6} + 21x^{5} - 38x^{4} + 46x^{3} - 32x^{2} + 24x - 8$$ x^9 - 2*x^8 + 4*x^7 - 8*x^6 + 21*x^5 - 38*x^4 + 46*x^3 - 32*x^2 + 24*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:
 $r_{ 1 }$ $=$ $$98 + 215\cdot 227 + 132\cdot 227^{2} + 20\cdot 227^{3} + 126\cdot 227^{4} + 108\cdot 227^{5} + 213\cdot 227^{6} + 44\cdot 227^{7} + 207\cdot 227^{8} + 224\cdot 227^{9} +O(227^{10})$$ 98 + 215*227 + 132*227^2 + 20*227^3 + 126*227^4 + 108*227^5 + 213*227^6 + 44*227^7 + 207*227^8 + 224*227^9+O(227^10) $r_{ 2 }$ $=$ $$116 + 190\cdot 227 + 194\cdot 227^{2} + 44\cdot 227^{3} + 181\cdot 227^{4} + 78\cdot 227^{5} + 113\cdot 227^{6} + 194\cdot 227^{7} + 6\cdot 227^{8} + 70\cdot 227^{9} +O(227^{10})$$ 116 + 190*227 + 194*227^2 + 44*227^3 + 181*227^4 + 78*227^5 + 113*227^6 + 194*227^7 + 6*227^8 + 70*227^9+O(227^10) $r_{ 3 }$ $=$ $$146 + 67\cdot 227 + 112\cdot 227^{2} + 4\cdot 227^{3} + 68\cdot 227^{4} + 21\cdot 227^{5} + 161\cdot 227^{6} + 7\cdot 227^{7} + 115\cdot 227^{8} + 30\cdot 227^{9} +O(227^{10})$$ 146 + 67*227 + 112*227^2 + 4*227^3 + 68*227^4 + 21*227^5 + 161*227^6 + 7*227^7 + 115*227^8 + 30*227^9+O(227^10) $r_{ 4 }$ $=$ $$72 a^{2} + 148 a + 50 + \left(188 a^{2} + 16 a + 212\right)\cdot 227 + \left(29 a^{2} + 176 a + 157\right)\cdot 227^{2} + \left(190 a^{2} + 115 a + 93\right)\cdot 227^{3} + \left(48 a^{2} + 199 a + 104\right)\cdot 227^{4} + \left(8 a^{2} + 221 a + 218\right)\cdot 227^{5} + \left(186 a^{2} + 115 a + 83\right)\cdot 227^{6} + \left(211 a^{2} + 6 a + 200\right)\cdot 227^{7} + \left(148 a^{2} + 225 a + 64\right)\cdot 227^{8} + \left(90 a^{2} + 46 a + 179\right)\cdot 227^{9} +O(227^{10})$$ 72*a^2 + 148*a + 50 + (188*a^2 + 16*a + 212)*227 + (29*a^2 + 176*a + 157)*227^2 + (190*a^2 + 115*a + 93)*227^3 + (48*a^2 + 199*a + 104)*227^4 + (8*a^2 + 221*a + 218)*227^5 + (186*a^2 + 115*a + 83)*227^6 + (211*a^2 + 6*a + 200)*227^7 + (148*a^2 + 225*a + 64)*227^8 + (90*a^2 + 46*a + 179)*227^9+O(227^10) $r_{ 5 }$ $=$ $$96 a^{2} + 123 a + 206 + \left(49 a^{2} + 166 a + 173\right)\cdot 227 + \left(174 a^{2} + 78 a + 118\right)\cdot 227^{2} + \left(120 a^{2} + 144 a + 70\right)\cdot 227^{3} + \left(150 a^{2} + 137 a + 36\right)\cdot 227^{4} + \left(100 a^{2} + 214 a + 84\right)\cdot 227^{5} + \left(57 a^{2} + 96 a + 2\right)\cdot 227^{6} + \left(133 a^{2} + 18 a + 26\right)\cdot 227^{7} + \left(31 a^{2} + 150 a + 66\right)\cdot 227^{8} + \left(183 a^{2} + 49 a + 77\right)\cdot 227^{9} +O(227^{10})$$ 96*a^2 + 123*a + 206 + (49*a^2 + 166*a + 173)*227 + (174*a^2 + 78*a + 118)*227^2 + (120*a^2 + 144*a + 70)*227^3 + (150*a^2 + 137*a + 36)*227^4 + (100*a^2 + 214*a + 84)*227^5 + (57*a^2 + 96*a + 2)*227^6 + (133*a^2 + 18*a + 26)*227^7 + (31*a^2 + 150*a + 66)*227^8 + (183*a^2 + 49*a + 77)*227^9+O(227^10) $r_{ 6 }$ $=$ $$140 a^{2} + 173 a + 189 + \left(174 a^{2} + 154 a + 113\right)\cdot 227 + \left(132 a^{2} + 171 a + 63\right)\cdot 227^{2} + \left(221 a^{2} + 155 a + 129\right)\cdot 227^{3} + \left(201 a^{2} + 147 a + 180\right)\cdot 227^{4} + \left(31 a^{2} + 149 a + 143\right)\cdot 227^{5} + \left(226 a^{2} + 52 a + 151\right)\cdot 227^{6} + \left(63 a^{2} + 69 a + 160\right)\cdot 227^{7} + \left(44 a^{2} + 21 a + 158\right)\cdot 227^{8} + \left(165 a^{2} + 24 a + 204\right)\cdot 227^{9} +O(227^{10})$$ 140*a^2 + 173*a + 189 + (174*a^2 + 154*a + 113)*227 + (132*a^2 + 171*a + 63)*227^2 + (221*a^2 + 155*a + 129)*227^3 + (201*a^2 + 147*a + 180)*227^4 + (31*a^2 + 149*a + 143)*227^5 + (226*a^2 + 52*a + 151)*227^6 + (63*a^2 + 69*a + 160)*227^7 + (44*a^2 + 21*a + 158)*227^8 + (165*a^2 + 24*a + 204)*227^9+O(227^10) $r_{ 7 }$ $=$ $$168 a^{2} + 168 a + 178 + \left(146 a^{2} + 76 a + 156\right)\cdot 227 + \left(120 a^{2} + 216 a + 127\right)\cdot 227^{2} + \left(134 a^{2} + 11 a + 19\right)\cdot 227^{3} + \left(166 a^{2} + 42 a + 110\right)\cdot 227^{4} + \left(124 a^{2} + 195 a + 222\right)\cdot 227^{5} + \left(162 a^{2} + 47 a + 203\right)\cdot 227^{6} + \left(66 a^{2} + 16 a + 6\right)\cdot 227^{7} + \left(41 a^{2} + 5 a + 224\right)\cdot 227^{8} + \left(43 a^{2} + 83 a + 115\right)\cdot 227^{9} +O(227^{10})$$ 168*a^2 + 168*a + 178 + (146*a^2 + 76*a + 156)*227 + (120*a^2 + 216*a + 127)*227^2 + (134*a^2 + 11*a + 19)*227^3 + (166*a^2 + 42*a + 110)*227^4 + (124*a^2 + 195*a + 222)*227^5 + (162*a^2 + 47*a + 203)*227^6 + (66*a^2 + 16*a + 6)*227^7 + (41*a^2 + 5*a + 224)*227^8 + (43*a^2 + 83*a + 115)*227^9+O(227^10) $r_{ 8 }$ $=$ $$214 a^{2} + 138 a + 88 + \left(118 a^{2} + 133 a + 195\right)\cdot 227 + \left(76 a^{2} + 61 a + 68\right)\cdot 227^{2} + \left(129 a^{2} + 99 a + 88\right)\cdot 227^{3} + \left(11 a^{2} + 212 a + 130\right)\cdot 227^{4} + \left(94 a^{2} + 36 a + 181\right)\cdot 227^{5} + \left(105 a^{2} + 63 a + 127\right)\cdot 227^{6} + \left(175 a^{2} + 204 a\right)\cdot 227^{7} + \left(36 a^{2} + 223 a + 218\right)\cdot 227^{8} + \left(93 a^{2} + 96 a + 106\right)\cdot 227^{9} +O(227^{10})$$ 214*a^2 + 138*a + 88 + (118*a^2 + 133*a + 195)*227 + (76*a^2 + 61*a + 68)*227^2 + (129*a^2 + 99*a + 88)*227^3 + (11*a^2 + 212*a + 130)*227^4 + (94*a^2 + 36*a + 181)*227^5 + (105*a^2 + 63*a + 127)*227^6 + (175*a^2 + 204*a)*227^7 + (36*a^2 + 223*a + 218)*227^8 + (93*a^2 + 96*a + 106)*227^9+O(227^10) $r_{ 9 }$ $=$ $$218 a^{2} + 158 a + 66 + \left(2 a^{2} + 132 a + 36\right)\cdot 227 + \left(147 a^{2} + 203 a + 158\right)\cdot 227^{2} + \left(111 a^{2} + 153 a + 209\right)\cdot 227^{3} + \left(101 a^{2} + 168 a + 197\right)\cdot 227^{4} + \left(94 a^{2} + 89 a + 75\right)\cdot 227^{5} + \left(170 a^{2} + 77 a + 77\right)\cdot 227^{6} + \left(29 a^{2} + 139 a + 39\right)\cdot 227^{7} + \left(151 a^{2} + 55 a + 74\right)\cdot 227^{8} + \left(105 a^{2} + 153 a + 125\right)\cdot 227^{9} +O(227^{10})$$ 218*a^2 + 158*a + 66 + (2*a^2 + 132*a + 36)*227 + (147*a^2 + 203*a + 158)*227^2 + (111*a^2 + 153*a + 209)*227^3 + (101*a^2 + 168*a + 197)*227^4 + (94*a^2 + 89*a + 75)*227^5 + (170*a^2 + 77*a + 77)*227^6 + (29*a^2 + 139*a + 39)*227^7 + (151*a^2 + 55*a + 74)*227^8 + (105*a^2 + 153*a + 125)*227^9+O(227^10)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.