Properties

Label 8.9120059001.12t213.a
Dimension $8$
Group $S_3\wr S_3$
Conductor $9120059001$
Indicator $1$

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Basic invariants

Dimension:$8$
Group:$S_3\wr S_3$
Conductor:\(9120059001\)\(\medspace = 3^{12} \cdot 131^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 9.1.32257648686537.1
Galois orbit size: $1$
Smallest permutation container: 12T213
Parity: even
Projective image: $S_3\wr S_3$
Projective field: Galois closure of 9.1.32257648686537.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 151 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 151 }$: \( x^{3} + x + 145 \) Copy content Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 10 a^{2} + 8 a + 134 + \left(54 a^{2} + 122 a + 140\right)\cdot 151 + \left(139 a^{2} + 52 a + 54\right)\cdot 151^{2} + \left(38 a^{2} + 12 a + 86\right)\cdot 151^{3} + \left(59 a^{2} + 45 a + 4\right)\cdot 151^{4} + \left(3 a^{2} + 109 a + 1\right)\cdot 151^{5} + \left(143 a^{2} + 16 a + 25\right)\cdot 151^{6} + \left(96 a^{2} + 19 a + 77\right)\cdot 151^{7} + \left(68 a^{2} + 97 a + 129\right)\cdot 151^{8} + \left(19 a^{2} + 93 a + 27\right)\cdot 151^{9} +O(151^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 15 a^{2} + 38 a + 77 + \left(94 a^{2} + 73 a + 89\right)\cdot 151 + \left(38 a^{2} + 42 a + 40\right)\cdot 151^{2} + \left(66 a^{2} + 52 a + 44\right)\cdot 151^{3} + \left(74 a^{2} + 130 a + 102\right)\cdot 151^{4} + \left(132 a^{2} + 134 a + 124\right)\cdot 151^{5} + \left(8 a^{2} + 113 a + 6\right)\cdot 151^{6} + \left(11 a^{2} + 130 a + 21\right)\cdot 151^{7} + \left(41 a^{2} + 109 a + 51\right)\cdot 151^{8} + \left(133 a^{2} + 93 a + 21\right)\cdot 151^{9} +O(151^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 18 a^{2} + 105 a + 89 + \left(8 a^{2} + 109 a + 9\right)\cdot 151 + \left(80 a^{2} + 42 a + 116\right)\cdot 151^{2} + \left(28 a^{2} + 39 a + 129\right)\cdot 151^{3} + \left(130 a^{2} + 123 a + 51\right)\cdot 151^{4} + \left(22 a^{2} + 142 a + 64\right)\cdot 151^{5} + \left(127 a^{2} + 37 a + 14\right)\cdot 151^{6} + \left(82 a^{2} + 110 a + 118\right)\cdot 151^{7} + \left(38 a^{2} + 77 a + 8\right)\cdot 151^{8} + \left(4 a^{2} + 100 a + 68\right)\cdot 151^{9} +O(151^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 28 a^{2} + 116 a + 77 + \left(63 a^{2} + 7 a + 61\right)\cdot 151 + \left(49 a^{2} + 48 a + 106\right)\cdot 151^{2} + \left(29 a^{2} + 88 a + 59\right)\cdot 151^{3} + \left(2 a^{2} + 18 a + 84\right)\cdot 151^{4} + \left(2 a^{2} + 121 a + 16\right)\cdot 151^{5} + \left(144 a^{2} + 28 a + 115\right)\cdot 151^{6} + \left(67 a^{2} + 58 a + 119\right)\cdot 151^{7} + \left(102 a^{2} + 130 a + 111\right)\cdot 151^{8} + \left(61 a^{2} + 46 a + 93\right)\cdot 151^{9} +O(151^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 50 a^{2} + 117 a + 50 + \left(104 a^{2} + 38 a + 96\right)\cdot 151 + \left(56 a^{2} + 31 a + 52\right)\cdot 151^{2} + \left(131 a^{2} + 131 a + 37\right)\cdot 151^{3} + \left(102 a^{2} + 126 a + 121\right)\cdot 151^{4} + \left(108 a^{2} + 21 a + 108\right)\cdot 151^{5} + \left(11 a^{2} + 118 a + 8\right)\cdot 151^{6} + \left(39 a^{2} + 143 a + 90\right)\cdot 151^{7} + \left(111 a^{2} + 128 a + 47\right)\cdot 151^{8} + \left(88 a^{2} + 150 a + 92\right)\cdot 151^{9} +O(151^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 86 a^{2} + 147 a + 74 + \left(103 a^{2} + 38 a + 45\right)\cdot 151 + \left(55 a^{2} + 77 a + 102\right)\cdot 151^{2} + \left(104 a^{2} + 118 a + 69\right)\cdot 151^{3} + \left(124 a^{2} + 44 a + 85\right)\cdot 151^{4} + \left(60 a^{2} + 145 a + 26\right)\cdot 151^{5} + \left(130 a^{2} + 69 a + 138\right)\cdot 151^{6} + \left(100 a^{2} + 27 a + 80\right)\cdot 151^{7} + \left(149 a^{2} + 63 a + 123\right)\cdot 151^{8} + \left(79 a^{2} + 57 a + 136\right)\cdot 151^{9} +O(151^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 123 a^{2} + 38 a + 8 + \left(88 a^{2} + 70 a + 13\right)\cdot 151 + \left(82 a^{2} + 55 a + 17\right)\cdot 151^{2} + \left(83 a^{2} + 99 a + 116\right)\cdot 151^{3} + \left(112 a^{2} + 133 a + 140\right)\cdot 151^{4} + \left(124 a^{2} + 49 a + 81\right)\cdot 151^{5} + \left(31 a^{2} + 96 a + 51\right)\cdot 151^{6} + \left(122 a^{2} + 21 a + 144\right)\cdot 151^{7} + \left(43 a^{2} + 127 a + 112\right)\cdot 151^{8} + \left(127 a^{2} + 107 a + 99\right)\cdot 151^{9} +O(151^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 132 a^{2} + 111 a + 96 + \left(113 a^{2} + 13 a + 145\right)\cdot 151 + \left(48 a^{2} + 42 a + 105\right)\cdot 151^{2} + \left(117 a^{2} + 98 a + 17\right)\cdot 151^{3} + \left(71 a^{2} + 77 a + 30\right)\cdot 151^{4} + \left(148 a^{2} + 14 a + 114\right)\cdot 151^{5} + \left(81 a^{2} + 70 a + 73\right)\cdot 151^{6} + \left(137 a^{2} + 52 a + 65\right)\cdot 151^{7} + \left(2 a^{2} + 79 a + 45\right)\cdot 151^{8} + \left(34 a^{2} + 91 a + 75\right)\cdot 151^{9} +O(151^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 142 a^{2} + 75 a + 2 + \left(124 a^{2} + 129 a + 2\right)\cdot 151 + \left(52 a^{2} + 60 a + 8\right)\cdot 151^{2} + \left(4 a^{2} + 115 a + 43\right)\cdot 151^{3} + \left(77 a^{2} + 54 a + 134\right)\cdot 151^{4} + \left(15 a + 65\right)\cdot 151^{5} + \left(76 a^{2} + 52 a + 19\right)\cdot 151^{6} + \left(96 a^{2} + 40 a + 38\right)\cdot 151^{7} + \left(45 a^{2} + 92 a + 124\right)\cdot 151^{8} + \left(55 a^{2} + 12 a + 139\right)\cdot 151^{9} +O(151^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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