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 $1$

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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)$$=$ \( x^{9} - 6x^{6} - 9x^{5} - 18x^{4} - 15x^{3} - 18x^{2} - 9x - 8 \) Copy content Toggle raw display .

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

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 227 }$: \( x^{3} + 2x + 225 \) Copy content Toggle raw display

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

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

SizeOrderAction 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.