Properties

Label 12.655...328.18t315.a.a
Dimension $12$
Group $S_3\wr S_3$
Conductor $6.558\times 10^{16}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $12$
Group: $S_3\wr S_3$
Conductor: \(65577574924899328\)\(\medspace = 2^{10} \cdot 31^{4} \cdot 37^{5}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 9.3.14873341696.1
Galois orbit size: $1$
Smallest permutation container: 18T315
Parity: even
Determinant: 1.37.2t1.a.a
Projective image: $C_3^3.S_4.C_2$
Projective stem field: Galois closure of 9.3.14873341696.1

Defining polynomial

$f(x)$$=$ \( x^{9} - 3x^{8} + 4x^{7} - 2x^{6} + 2x^{5} - 2x^{4} - 2x^{3} + 8x^{2} - 3x - 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 157 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 157 }$: \( x^{3} + x + 152 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( a^{2} + 58 a + 46 + \left(155 a^{2} + 57 a + 75\right)\cdot 157 + \left(23 a + 81\right)\cdot 157^{2} + \left(6 a^{2} + 74 a + 10\right)\cdot 157^{3} + \left(52 a^{2} + 6 a + 125\right)\cdot 157^{4} + \left(95 a^{2} + 114 a + 16\right)\cdot 157^{5} + \left(110 a^{2} + 140 a + 23\right)\cdot 157^{6} + \left(86 a^{2} + 43 a + 100\right)\cdot 157^{7} + \left(97 a + 127\right)\cdot 157^{8} + \left(16 a^{2} + 123 a + 59\right)\cdot 157^{9} +O(157^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 24 a^{2} + 103 a + 85 + \left(93 a^{2} + 22 a + 133\right)\cdot 157 + \left(37 a^{2} + 6 a + 34\right)\cdot 157^{2} + \left(62 a^{2} + 86 a + 109\right)\cdot 157^{3} + \left(131 a^{2} + 104 a + 98\right)\cdot 157^{4} + \left(118 a^{2} + 70 a + 114\right)\cdot 157^{5} + \left(36 a^{2} + 50 a + 142\right)\cdot 157^{6} + \left(96 a^{2} + 106 a + 88\right)\cdot 157^{7} + \left(113 a^{2} + 121 a + 104\right)\cdot 157^{8} + \left(116 a^{2} + 136 a + 41\right)\cdot 157^{9} +O(157^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 37 a^{2} + 44 a + 70 + \left(18 a^{2} + 116 a + 36\right)\cdot 157 + \left(91 a^{2} + 27 a + 89\right)\cdot 157^{2} + \left(126 a^{2} + 57 a + 38\right)\cdot 157^{3} + \left(30 a^{2} + 82 a + 6\right)\cdot 157^{4} + \left(138 a^{2} + 88 a + 150\right)\cdot 157^{5} + \left(18 a^{2} + 101 a + 118\right)\cdot 157^{6} + \left(92 a^{2} + 124 a + 103\right)\cdot 157^{7} + \left(29 a^{2} + 96 a + 94\right)\cdot 157^{8} + \left(88 a^{2} + 44 a + 55\right)\cdot 157^{9} +O(157^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 55 a^{2} + 99 a + 28 + \left(59 a^{2} + 32 a + 153\right)\cdot 157 + \left(119 a^{2} + 156 a + 145\right)\cdot 157^{2} + \left(125 a^{2} + 62 a + 61\right)\cdot 157^{3} + \left(39 a^{2} + 58 a + 134\right)\cdot 157^{4} + \left(39 a^{2} + 136 a + 89\right)\cdot 157^{5} + \left(56 a^{2} + 80 a + 74\right)\cdot 157^{6} + \left(26 a^{2} + 83 a + 107\right)\cdot 157^{7} + \left(63 a^{2} + 31 a + 147\right)\cdot 157^{8} + \left(2 a^{2} + 111 a + 40\right)\cdot 157^{9} +O(157^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 105 a^{2} + 88 a + 9 + \left(62 a^{2} + 89 a + 103\right)\cdot 157 + \left(128 a^{2} + 74 a + 99\right)\cdot 157^{2} + \left(78 a^{2} + 147 a + 30\right)\cdot 157^{3} + \left(5 a^{2} + 64 a + 59\right)\cdot 157^{4} + \left(99 a^{2} + 45 a + 77\right)\cdot 157^{5} + \left(10 a^{2} + 92 a + 96\right)\cdot 157^{6} + \left(44 a^{2} + 12 a + 14\right)\cdot 157^{7} + \left(145 a^{2} + 111 a + 150\right)\cdot 157^{8} + \left(71 a^{2} + 77 a + 34\right)\cdot 157^{9} +O(157^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 119 a^{2} + 55 a + 20 + \left(140 a^{2} + 140 a + 118\right)\cdot 157 + \left(64 a^{2} + 105 a + 71\right)\cdot 157^{2} + \left(24 a^{2} + 25 a + 127\right)\cdot 157^{3} + \left(74 a^{2} + 68 a + 139\right)\cdot 157^{4} + \left(80 a^{2} + 111 a + 6\right)\cdot 157^{5} + \left(27 a^{2} + 71 a + 20\right)\cdot 157^{6} + \left(135 a^{2} + 145 a + 80\right)\cdot 157^{7} + \left(126 a^{2} + 119 a + 2\right)\cdot 157^{8} + \left(52 a^{2} + 145 a + 32\right)\cdot 157^{9} +O(157^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 140 a^{2} + 42 a + 110 + \left(105 a^{2} + 124 a + 89\right)\cdot 157 + \left(20 a^{2} + 25 a + 23\right)\cdot 157^{2} + \left(77 a^{2} + 40 a + 119\right)\cdot 157^{3} + \left(27 a^{2} + 68 a + 81\right)\cdot 157^{4} + \left(29 a^{2} + 3 a + 2\right)\cdot 157^{5} + \left(57 a^{2} + 76 a + 104\right)\cdot 157^{6} + \left(138 a^{2} + 151 a + 64\right)\cdot 157^{7} + \left(128 a^{2} + 47 a + 62\right)\cdot 157^{8} + \left(87 a^{2} + 36 a + 22\right)\cdot 157^{9} +O(157^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 150 a^{2} + 12 a + 12 + \left(114 a^{2} + 10 a + 148\right)\cdot 157 + \left(98 a^{2} + 125 a + 127\right)\cdot 157^{2} + \left(17 a^{2} + 30 a + 131\right)\cdot 157^{3} + \left(155 a^{2} + 141 a + 9\right)\cdot 157^{4} + \left(8 a^{2} + 82 a + 146\right)\cdot 157^{5} + \left(63 a^{2} + 30 a + 107\right)\cdot 157^{6} + \left(79 a^{2} + 56 a + 77\right)\cdot 157^{7} + \left(71 a^{2} + 144 a + 76\right)\cdot 157^{8} + \left(109 a^{2} + 140 a + 141\right)\cdot 157^{9} +O(157^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 154 a^{2} + 127 a + 94 + \left(34 a^{2} + 34 a + 84\right)\cdot 157 + \left(66 a^{2} + 83 a + 110\right)\cdot 157^{2} + \left(109 a^{2} + 103 a + 155\right)\cdot 157^{3} + \left(111 a^{2} + 33 a + 129\right)\cdot 157^{4} + \left(18 a^{2} + 132 a + 23\right)\cdot 157^{5} + \left(90 a^{2} + 140 a + 97\right)\cdot 157^{6} + \left(86 a^{2} + 60 a + 147\right)\cdot 157^{7} + \left(105 a^{2} + 14 a + 18\right)\cdot 157^{8} + \left(82 a^{2} + 125 a + 42\right)\cdot 157^{9} +O(157^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.