Properties

Label 12.369...344.18t315.a.a
Dimension $12$
Group $S_3\wr S_3$
Conductor $3.692\times 10^{16}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $12$
Group: $S_3\wr S_3$
Conductor: \(36922207099409344\)\(\medspace = 2^{6} \cdot 31^{5} \cdot 67^{4}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 9.1.127743808.1
Galois orbit size: $1$
Smallest permutation container: 18T315
Parity: odd
Determinant: 1.31.2t1.a.a
Projective image: $C_3^3.S_4.C_2$
Projective stem field: Galois closure of 9.1.127743808.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 149 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 149 }$: \( x^{3} + 3x + 147 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 33 + 26\cdot 149 + 103\cdot 149^{2} + 93\cdot 149^{3} + 50\cdot 149^{4} + 30\cdot 149^{5} + 89\cdot 149^{6} + 18\cdot 149^{7} + 111\cdot 149^{8} + 129\cdot 149^{9} +O(149^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 53 + 81\cdot 149 + 101\cdot 149^{2} + 143\cdot 149^{3} + 74\cdot 149^{4} + 7\cdot 149^{5} + 132\cdot 149^{6} + 139\cdot 149^{7} + 146\cdot 149^{8} + 98\cdot 149^{9} +O(149^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 91 + 43\cdot 149 + 58\cdot 149^{2} + 23\cdot 149^{3} + 107\cdot 149^{4} + 110\cdot 149^{5} + 60\cdot 149^{6} + 72\cdot 149^{7} + 91\cdot 149^{8} + 31\cdot 149^{9} +O(149^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 7 a^{2} + 82 a + 107 + \left(99 a^{2} + 12 a + 9\right)\cdot 149 + \left(39 a^{2} + 14 a + 140\right)\cdot 149^{2} + \left(106 a^{2} + 46 a + 134\right)\cdot 149^{3} + \left(43 a^{2} + 126 a + 15\right)\cdot 149^{4} + \left(58 a^{2} + 8 a + 22\right)\cdot 149^{5} + \left(117 a + 118\right)\cdot 149^{6} + \left(132 a^{2} + 23 a + 51\right)\cdot 149^{7} + \left(51 a^{2} + 86 a + 77\right)\cdot 149^{8} + \left(94 a^{2} + 33 a + 105\right)\cdot 149^{9} +O(149^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 55 a^{2} + 91 a + 54 + \left(121 a^{2} + 20 a + 54\right)\cdot 149 + \left(117 a^{2} + 61 a + 147\right)\cdot 149^{2} + \left(137 a^{2} + 120 a + 48\right)\cdot 149^{3} + \left(13 a^{2} + 127 a + 105\right)\cdot 149^{4} + \left(49 a^{2} + 107 a + 3\right)\cdot 149^{5} + \left(87 a^{2} + 19 a + 143\right)\cdot 149^{6} + \left(64 a^{2} + 8 a + 65\right)\cdot 149^{7} + \left(77 a^{2} + 83 a + 128\right)\cdot 149^{8} + \left(66 a^{2} + 29 a + 49\right)\cdot 149^{9} +O(149^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 82 a^{2} + 105 a + 112 + \left(33 a^{2} + 118 a + 105\right)\cdot 149 + \left(123 a^{2} + 6 a + 147\right)\cdot 149^{2} + \left(22 a^{2} + 55 a + 85\right)\cdot 149^{3} + \left(39 a^{2} + 117 a + 22\right)\cdot 149^{4} + \left(139 a^{2} + 120 a + 75\right)\cdot 149^{5} + \left(40 a^{2} + 117 a + 19\right)\cdot 149^{6} + \left(19 a^{2} + 29 a + 124\right)\cdot 149^{7} + \left(93 a^{2} + 33 a + 145\right)\cdot 149^{8} + \left(70 a^{2} + 73 a + 87\right)\cdot 149^{9} +O(149^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 87 a^{2} + 125 a + 118 + \left(77 a^{2} + 115 a + 115\right)\cdot 149 + \left(140 a^{2} + 73 a + 43\right)\cdot 149^{2} + \left(53 a^{2} + 131 a + 30\right)\cdot 149^{3} + \left(91 a^{2} + 43 a + 111\right)\cdot 149^{4} + \left(41 a^{2} + 32 a + 137\right)\cdot 149^{5} + \left(61 a^{2} + 12 a + 90\right)\cdot 149^{6} + \left(101 a^{2} + 117 a + 139\right)\cdot 149^{7} + \left(19 a^{2} + 128 a + 12\right)\cdot 149^{8} + \left(137 a^{2} + 85 a + 42\right)\cdot 149^{9} +O(149^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 99 a^{2} + 46 a + 146 + \left(98 a^{2} + 6 a + 86\right)\cdot 149 + \left(7 a^{2} + 48 a + 65\right)\cdot 149^{2} + \left(124 a^{2} + 101 a + 139\right)\cdot 149^{3} + \left(114 a^{2} + 58 a + 24\right)\cdot 149^{4} + \left(66 a^{2} + 111 a + 79\right)\cdot 149^{5} + \left(108 a^{2} + 31 a + 5\right)\cdot 149^{6} + \left(41 a^{2} + 5 a + 20\right)\cdot 149^{7} + \left(23 a^{2} + 120 a + 6\right)\cdot 149^{8} + \left(129 a^{2} + 133 a + 56\right)\cdot 149^{9} +O(149^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 117 a^{2} + 147 a + 33 + \left(16 a^{2} + 23 a + 72\right)\cdot 149 + \left(18 a^{2} + 94 a + 86\right)\cdot 149^{2} + \left(2 a^{2} + 141 a + 44\right)\cdot 149^{3} + \left(144 a^{2} + 121 a + 83\right)\cdot 149^{4} + \left(91 a^{2} + 65 a + 129\right)\cdot 149^{5} + \left(148 a^{2} + 148 a + 85\right)\cdot 149^{6} + \left(87 a^{2} + 113 a + 112\right)\cdot 149^{7} + \left(32 a^{2} + 144 a + 24\right)\cdot 149^{8} + \left(98 a^{2} + 90 a + 143\right)\cdot 149^{9} +O(149^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.