Properties

Label 6.253...096.18t300.a.a
Dimension $6$
Group $S_3\wr S_3$
Conductor $2.540\times 10^{14}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $6$
Group: $S_3\wr S_3$
Conductor: \(253958135812096\)\(\medspace = 2^{12} \cdot 499^{4} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 9.1.7952095936.1
Galois orbit size: $1$
Smallest permutation container: 18T300
Parity: odd
Determinant: 1.4.2t1.a.a
Projective image: $S_3\wr S_3$
Projective stem field: Galois closure of 9.1.7952095936.1

Defining polynomial

$f(x)$$=$ \( x^{9} - 2x^{8} + 2x^{7} + 3x^{6} - 2x^{5} - 10x^{4} + 5x^{3} + 14x^{2} + 6x + 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 109 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 109 }$: \( x^{3} + x + 103 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 43 a^{2} + 62 a + 23 + \left(41 a^{2} + 24 a + 87\right)\cdot 109 + \left(77 a^{2} + 31 a + 47\right)\cdot 109^{2} + \left(84 a^{2} + 20 a + 65\right)\cdot 109^{3} + \left(64 a^{2} + 69 a + 78\right)\cdot 109^{4} + \left(100 a^{2} + 102 a + 106\right)\cdot 109^{5} + \left(60 a^{2} + 79 a + 64\right)\cdot 109^{6} + \left(86 a^{2} + 44 a + 11\right)\cdot 109^{7} + \left(90 a^{2} + 17 a + 57\right)\cdot 109^{8} + \left(96 a^{2} + 33 a + 31\right)\cdot 109^{9} +O(109^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 48 a^{2} + 45 a + 66 + \left(92 a^{2} + 69 a + 108\right)\cdot 109 + \left(105 a^{2} + 15 a + 72\right)\cdot 109^{2} + \left(5 a^{2} + 21 a + 31\right)\cdot 109^{3} + \left(75 a^{2} + 99 a + 80\right)\cdot 109^{4} + \left(67 a^{2} + 47 a + 86\right)\cdot 109^{5} + \left(25 a^{2} + 66 a + 6\right)\cdot 109^{6} + \left(53 a^{2} + 93 a + 33\right)\cdot 109^{7} + \left(87 a^{2} + 42 a + 79\right)\cdot 109^{8} + \left(3 a^{2} + 13 a + 21\right)\cdot 109^{9} +O(109^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 49 a^{2} + 52 a + 5 + \left(93 a^{2} + 48 a + 101\right)\cdot 109 + \left(13 a^{2} + 73 a + 46\right)\cdot 109^{2} + \left(62 a^{2} + 28 a + 77\right)\cdot 109^{3} + \left(54 a^{2} + 77 a + 79\right)\cdot 109^{4} + \left(38 a^{2} + 104 a + 89\right)\cdot 109^{5} + \left(61 a^{2} + 5 a + 26\right)\cdot 109^{6} + \left(93 a^{2} + 78 a + 38\right)\cdot 109^{7} + \left(26 a^{2} + 31 a\right)\cdot 109^{8} + \left(10 a^{2} + 33 a + 57\right)\cdot 109^{9} +O(109^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 70 a^{2} + 19 a + 19 + \left(96 a^{2} + 75 a + 103\right)\cdot 109 + \left(104 a^{2} + 51 a + 34\right)\cdot 109^{2} + \left(73 a^{2} + 64 a + 85\right)\cdot 109^{3} + \left(4 a^{2} + 45 a + 82\right)\cdot 109^{4} + \left(15 a^{2} + 41 a + 37\right)\cdot 109^{5} + \left(11 a^{2} + 57 a + 102\right)\cdot 109^{6} + \left(39 a^{2} + 104 a + 1\right)\cdot 109^{7} + \left(44 a^{2} + 38 a + 12\right)\cdot 109^{8} + \left(37 a^{2} + 57 a + 75\right)\cdot 109^{9} +O(109^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 78 a^{2} + 3 a + 86 + \left(81 a^{2} + 43 a + 28\right)\cdot 109 + \left(91 a^{2} + 42 a + 27\right)\cdot 109^{2} + \left(24 a^{2} + 101 a + 44\right)\cdot 109^{3} + \left(31 a^{2} + 68 a + 87\right)\cdot 109^{4} + \left(29 a^{2} + 70 a + 24\right)\cdot 109^{5} + \left(107 a^{2} + 64 a + 61\right)\cdot 109^{6} + \left(56 a^{2} + 34 a + 35\right)\cdot 109^{7} + \left(32 a^{2} + 39 a + 6\right)\cdot 109^{8} + \left(29 a^{2} + 29 a + 75\right)\cdot 109^{9} +O(109^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 80 a^{2} + 30 a + 84 + \left(74 a^{2} + 15 a + 36\right)\cdot 109 + \left(73 a^{2} + 55 a + 45\right)\cdot 109^{2} + \left(12 a^{2} + 93 a + 17\right)\cdot 109^{3} + \left(94 a^{2} + 45 a + 98\right)\cdot 109^{4} + \left(68 a^{2} + 18 a + 12\right)\cdot 109^{5} + \left(80 a^{2} + 90 a + 78\right)\cdot 109^{6} + \left(51 a^{2} + 32 a + 24\right)\cdot 109^{7} + \left(75 a^{2} + 83 a + 83\right)\cdot 109^{8} + \left(77 a^{2} + 7 a + 18\right)\cdot 109^{9} +O(109^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 92 a^{2} + 61 a + 59 + \left(43 a^{2} + 105 a + 3\right)\cdot 109 + \left(20 a^{2} + 50 a + 16\right)\cdot 109^{2} + \left(78 a^{2} + 95 a + 7\right)\cdot 109^{3} + \left(2 a^{2} + 49 a + 32\right)\cdot 109^{4} + \left(12 a^{2} + 99 a + 13\right)\cdot 109^{5} + \left(85 a^{2} + 86 a + 10\right)\cdot 109^{6} + \left(107 a^{2} + 89 a + 33\right)\cdot 109^{7} + \left(97 a^{2} + 26 a + 86\right)\cdot 109^{8} + \left(75 a^{2} + 66 a + 69\right)\cdot 109^{9} +O(109^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 95 a^{2} + 17 a + 94 + \left(101 a^{2} + 69 a + 54\right)\cdot 109 + \left(66 a^{2} + 22 a + 4\right)\cdot 109^{2} + \left(11 a^{2} + 104 a + 53\right)\cdot 109^{3} + \left(59 a^{2} + 102 a + 38\right)\cdot 109^{4} + \left(48 a^{2} + 96 a + 108\right)\cdot 109^{5} + \left(76 a^{2} + 47 a + 38\right)\cdot 109^{6} + \left(79 a^{2} + 31 a + 43\right)\cdot 109^{7} + \left(51 a^{2} + 8 a + 67\right)\cdot 109^{8} + \left(43 a^{2} + 68 a + 68\right)\cdot 109^{9} +O(109^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 99 a^{2} + 38 a + 2 + \left(27 a^{2} + 94 a + 21\right)\cdot 109 + \left(99 a^{2} + 92 a + 31\right)\cdot 109^{2} + \left(81 a^{2} + 15 a + 54\right)\cdot 109^{3} + \left(49 a^{2} + 95 a + 76\right)\cdot 109^{4} + \left(55 a^{2} + 71 a + 64\right)\cdot 109^{5} + \left(36 a^{2} + 45 a + 46\right)\cdot 109^{6} + \left(85 a^{2} + 35 a + 105\right)\cdot 109^{7} + \left(37 a^{2} + 38 a + 43\right)\cdot 109^{8} + \left(61 a^{2} + 18 a + 18\right)\cdot 109^{9} +O(109^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.