Properties

Label 8.11110846464.12t213.a
Dimension $8$
Group $S_3\wr S_3$
Conductor $11110846464$
Indicator $1$

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Basic invariants

Dimension:$8$
Group:$S_3\wr S_3$
Conductor:\(11110846464\)\(\medspace = 2^{12} \cdot 3^{6} \cdot 61^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 9.1.37653424128.1
Galois orbit size: $1$
Smallest permutation container: 12T213
Parity: even
Projective image: $S_3\wr S_3$
Projective field: Galois closure of 9.1.37653424128.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 181 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 181 }$: \( x^{3} + 6x + 179 \) Copy content Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 6 a^{2} + 7 a + 3 + \left(56 a^{2} + 107 a + 71\right)\cdot 181 + \left(63 a^{2} + 113 a + 55\right)\cdot 181^{2} + \left(8 a^{2} + 82 a + 40\right)\cdot 181^{3} + \left(171 a^{2} + 25 a + 50\right)\cdot 181^{4} + \left(151 a^{2} + 60 a + 179\right)\cdot 181^{5} + \left(145 a^{2} + 101 a + 15\right)\cdot 181^{6} + \left(37 a^{2} + 67 a + 16\right)\cdot 181^{7} + \left(142 a^{2} + 129 a + 21\right)\cdot 181^{8} + \left(53 a^{2} + 151 a + 58\right)\cdot 181^{9} +O(181^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 48 a^{2} + 163 a + 165 + \left(11 a^{2} + 143 a + 86\right)\cdot 181 + \left(6 a^{2} + 118 a + 175\right)\cdot 181^{2} + \left(168 a^{2} + 65 a + 101\right)\cdot 181^{3} + \left(78 a^{2} + 95 a + 124\right)\cdot 181^{4} + \left(85 a^{2} + 62 a + 161\right)\cdot 181^{5} + \left(139 a^{2} + 44 a + 129\right)\cdot 181^{6} + \left(18 a^{2} + 24 a + 152\right)\cdot 181^{7} + \left(81 a^{2} + 158 a + 18\right)\cdot 181^{8} + \left(153 a^{2} + 67 a + 155\right)\cdot 181^{9} +O(181^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 53 a^{2} + 33 a + 19 + \left(167 a^{2} + 19 a + 177\right)\cdot 181 + \left(166 a^{2} + 11 a + 110\right)\cdot 181^{2} + \left(7 a^{2} + 160 a + 172\right)\cdot 181^{3} + \left(26 a^{2} + 143 a + 144\right)\cdot 181^{4} + \left(173 a^{2} + 57 a + 154\right)\cdot 181^{5} + \left(136 a^{2} + 153 a + 34\right)\cdot 181^{6} + \left(4 a^{2} + 155 a + 16\right)\cdot 181^{7} + \left(45 a^{2} + 61 a + 68\right)\cdot 181^{8} + \left(37 a^{2} + 60 a + 161\right)\cdot 181^{9} +O(181^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 63 a^{2} + 83 a + 59 + \left(177 a^{2} + 55 a + 36\right)\cdot 181 + \left(113 a^{2} + 130 a + 80\right)\cdot 181^{2} + \left(14 a^{2} + 141 a + 18\right)\cdot 181^{3} + \left(42 a^{2} + 59 a + 28\right)\cdot 181^{4} + \left(30 a^{2} + 34 a + 126\right)\cdot 181^{5} + \left(58 a^{2} + 56 a + 81\right)\cdot 181^{6} + \left(125 a^{2} + 47 a + 136\right)\cdot 181^{7} + \left(18 a^{2} + 153 a + 143\right)\cdot 181^{8} + \left(56 a^{2} + 65 a + 55\right)\cdot 181^{9} +O(181^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 65 a^{2} + 65 a + 67 + \left(17 a^{2} + 106 a + 120\right)\cdot 181 + \left(81 a^{2} + 39 a + 129\right)\cdot 181^{2} + \left(158 a^{2} + 60 a + 50\right)\cdot 181^{3} + \left(112 a^{2} + 158 a + 130\right)\cdot 181^{4} + \left(158 a^{2} + 88 a + 96\right)\cdot 181^{5} + \left(166 a^{2} + 152 a + 154\right)\cdot 181^{6} + \left(50 a^{2} + 158 a + 19\right)\cdot 181^{7} + \left(117 a^{2} + 146 a + 176\right)\cdot 181^{8} + \left(87 a^{2} + 54 a\right)\cdot 181^{9} +O(181^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 66 a^{2} + 15 a + 62 + \left(100 a^{2} + 136 a + 67\right)\cdot 181 + \left(164 a^{2} + 21 a + 98\right)\cdot 181^{2} + \left(150 a^{2} + 34 a + 67\right)\cdot 181^{3} + \left(177 a^{2} + 75 a + 77\right)\cdot 181^{4} + \left(54 a^{2} + 41 a + 153\right)\cdot 181^{5} + \left(170 a^{2} + 43 a + 113\right)\cdot 181^{6} + \left(180 a^{2} + 110 a + 45\right)\cdot 181^{7} + \left(144 a^{2} + 101 a + 32\right)\cdot 181^{8} + \left(154 a^{2} + 179 a + 100\right)\cdot 181^{9} +O(181^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 66 a^{2} + 107 a + 56 + \left(87 a^{2} + 71 a + 29\right)\cdot 181 + \left(144 a^{2} + 52 a + 5\right)\cdot 181^{2} + \left(106 a^{2} + 76 a + 38\right)\cdot 181^{3} + \left(114 a^{2} + 68 a + 86\right)\cdot 181^{4} + \left(4 a^{2} + 9 a + 19\right)\cdot 181^{5} + \left(60 a^{2} + 24 a + 174\right)\cdot 181^{6} + \left(130 a^{2} + 173 a + 55\right)\cdot 181^{7} + \left(162 a^{2} + 20 a + 164\right)\cdot 181^{8} + \left(51 a^{2} + 154 a + 110\right)\cdot 181^{9} +O(181^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 67 a^{2} + 92 a + 60 + \left(82 a^{2} + 146 a + 9\right)\cdot 181 + \left(30 a^{2} + 9 a + 92\right)\cdot 181^{2} + \left(87 a^{2} + 39 a + 140\right)\cdot 181^{3} + \left(168 a^{2} + 17 a + 120\right)\cdot 181^{4} + \left(90 a^{2} + 109 a + 2\right)\cdot 181^{5} + \left(162 a^{2} + 112 a + 41\right)\cdot 181^{6} + \left(31 a^{2} + 164 a + 24\right)\cdot 181^{7} + \left(118 a^{2} + a + 167\right)\cdot 181^{8} + \left(156 a^{2} + 140 a + 167\right)\cdot 181^{9} +O(181^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 109 a^{2} + 159 a + 53 + \left(24 a^{2} + 118 a + 126\right)\cdot 181 + \left(134 a^{2} + 45 a + 157\right)\cdot 181^{2} + \left(21 a^{2} + 64 a + 93\right)\cdot 181^{3} + \left(13 a^{2} + 80 a + 142\right)\cdot 181^{4} + \left(155 a^{2} + 79 a + 10\right)\cdot 181^{5} + \left(45 a^{2} + 36 a + 159\right)\cdot 181^{6} + \left(143 a^{2} + 3 a + 75\right)\cdot 181^{7} + \left(74 a^{2} + 131 a + 113\right)\cdot 181^{8} + \left(153 a^{2} + 30 a + 94\right)\cdot 181^{9} +O(181^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 9 }$ Character values
$c1$
$1$ $1$ $()$ $8$
$9$ $2$ $(2,5)$ $0$
$18$ $2$ $(2,4)(5,6)(8,9)$ $4$
$27$ $2$ $(1,3)(2,5)(4,6)$ $0$
$27$ $2$ $(2,5)(4,6)$ $0$
$54$ $2$ $(1,4)(2,5)(3,6)(7,8)$ $0$
$6$ $3$ $(1,3,7)$ $-4$
$8$ $3$ $(1,3,7)(2,5,9)(4,6,8)$ $-1$
$12$ $3$ $(1,3,7)(4,6,8)$ $2$
$72$ $3$ $(1,2,4)(3,5,6)(7,9,8)$ $2$
$54$ $4$ $(2,6,5,4)(8,9)$ $0$
$162$ $4$ $(1,2,3,5)(6,8)(7,9)$ $0$
$36$ $6$ $(1,3,7)(2,4)(5,6)(8,9)$ $-2$
$36$ $6$ $(1,5,3,9,7,2)$ $-2$
$36$ $6$ $(1,3,7)(2,5)$ $0$
$36$ $6$ $(1,3,7)(2,5)(4,6,8)$ $0$
$54$ $6$ $(1,7,3)(2,5)(4,6)$ $0$
$72$ $6$ $(1,3,7)(2,4,5,6,9,8)$ $1$
$108$ $6$ $(1,6,3,8,7,4)(2,5)$ $0$
$216$ $6$ $(1,2,6,3,5,4)(7,9,8)$ $0$
$144$ $9$ $(1,5,6,3,9,8,7,2,4)$ $-1$
$108$ $12$ $(1,3,7)(2,6,5,4)(8,9)$ $0$
The blue line marks the conjugacy class containing complex conjugation.