Properties

Label 12.243...861.18t315.b.a
Dimension $12$
Group $S_3\wr S_3$
Conductor $2.440\times 10^{20}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $12$
Group: $S_3\wr S_3$
Conductor: \(243\!\cdots\!861\)\(\medspace = 3^{18} \cdot 229^{5} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 9.7.236372930487.1
Galois orbit size: $1$
Smallest permutation container: 18T315
Parity: even
Determinant: 1.229.2t1.a.a
Projective image: $S_3\wr S_3$
Projective stem field: Galois closure of 9.7.236372930487.1

Defining polynomial

$f(x)$$=$ \( x^{9} - 9x^{7} + 27x^{5} - 31x^{3} + 12x - 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 103 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 103 }$: \( x^{3} + 2x + 98 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 3 a^{2} + 2 a + 12 + \left(79 a^{2} + 92 a + 90\right)\cdot 103 + \left(25 a^{2} + 48 a + 73\right)\cdot 103^{2} + \left(26 a^{2} + 22 a + 39\right)\cdot 103^{3} + \left(3 a^{2} + 63 a + 47\right)\cdot 103^{4} + \left(65 a^{2} + 28 a + 86\right)\cdot 103^{5} + \left(40 a^{2} + 40 a + 96\right)\cdot 103^{6} + \left(44 a^{2} + 17 a + 84\right)\cdot 103^{7} + \left(42 a^{2} + 2 a + 91\right)\cdot 103^{8} + \left(33 a^{2} + 8 a + 79\right)\cdot 103^{9} +O(103^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 14 a^{2} + 100 a + 61 + \left(97 a^{2} + 23 a + 45\right)\cdot 103 + \left(41 a^{2} + 51 a + 95\right)\cdot 103^{2} + \left(17 a^{2} + 32 a + 27\right)\cdot 103^{3} + \left(14 a^{2} + 54 a + 96\right)\cdot 103^{4} + \left(73 a^{2} + 92 a + 62\right)\cdot 103^{5} + \left(55 a^{2} + 31 a + 82\right)\cdot 103^{6} + \left(29 a^{2} + 57 a + 30\right)\cdot 103^{7} + \left(76 a^{2} + 95 a + 68\right)\cdot 103^{8} + \left(66 a^{2} + 101 a + 55\right)\cdot 103^{9} +O(103^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 25 a^{2} + a + 6 + \left(76 a^{2} + 94 a + 97\right)\cdot 103 + \left(76 a^{2} + 25 a + 68\right)\cdot 103^{2} + \left(25 a^{2} + 24 a + 35\right)\cdot 103^{3} + \left(16 a^{2} + 10 a + 94\right)\cdot 103^{4} + \left(31 a^{2} + 3 a + 93\right)\cdot 103^{5} + \left(76 a^{2} + 44 a + 20\right)\cdot 103^{6} + \left(49 a^{2} + 5\right)\cdot 103^{7} + \left(62 a^{2} + 71 a + 47\right)\cdot 103^{8} + \left(69 a^{2} + 72 a + 81\right)\cdot 103^{9} +O(103^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 26 a^{2} + 72 a + 76 + \left(42 a^{2} + 34 a + 51\right)\cdot 103 + \left(46 a^{2} + 43 a + 28\right)\cdot 103^{2} + \left(56 a^{2} + 54 a + 42\right)\cdot 103^{3} + \left(16 a^{2} + 34 a + 60\right)\cdot 103^{4} + \left(90 a^{2} + 71 a + 69\right)\cdot 103^{5} + \left(29 a^{2} + 6 a + 27\right)\cdot 103^{6} + \left(5 a^{2} + 8 a + 83\right)\cdot 103^{7} + \left(91 a^{2} + 14 a + 50\right)\cdot 103^{8} + \left(45 a^{2} + 11 a + 15\right)\cdot 103^{9} +O(103^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 52 a^{2} + 30 a + 42 + \left(87 a^{2} + 77 a + 43\right)\cdot 103 + \left(82 a^{2} + 33 a + 8\right)\cdot 103^{2} + \left(20 a^{2} + 24 a + 29\right)\cdot 103^{3} + \left(70 a^{2} + 58 a + 63\right)\cdot 103^{4} + \left(84 a^{2} + 28 a + 96\right)\cdot 103^{5} + \left(99 a^{2} + 52 a + 17\right)\cdot 103^{6} + \left(47 a^{2} + 94 a + 37\right)\cdot 103^{7} + \left(52 a^{2} + 17 a + 102\right)\cdot 103^{8} + \left(90 a^{2} + 19 a + 74\right)\cdot 103^{9} +O(103^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 63 a^{2} + 34 a + 69 + \left(66 a^{2} + 44 a + 5\right)\cdot 103 + \left(14 a^{2} + 8 a + 82\right)\cdot 103^{2} + \left(29 a^{2} + 16 a + 32\right)\cdot 103^{3} + \left(72 a^{2} + 14 a + 49\right)\cdot 103^{4} + \left(42 a^{2} + 42 a + 73\right)\cdot 103^{5} + \left(17 a^{2} + 64 a + 95\right)\cdot 103^{6} + \left(68 a^{2} + 37 a + 91\right)\cdot 103^{7} + \left(38 a^{2} + 96 a + 86\right)\cdot 103^{8} + \left(93 a^{2} + 92 a + 31\right)\cdot 103^{9} +O(103^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 68 a^{2} + 72 a + 7 + \left(88 a^{2} + 38 a + 35\right)\cdot 103 + \left(87 a^{2} + 66 a + 42\right)\cdot 103^{2} + \left(22 a^{2} + 30 a + 24\right)\cdot 103^{3} + \left(50 a^{2} + 59 a + 54\right)\cdot 103^{4} + \left(53 a^{2} + 92 a + 53\right)\cdot 103^{5} + \left(99 a^{2} + 19 a + 33\right)\cdot 103^{6} + \left(25 a^{2} + 83 a + 1\right)\cdot 103^{7} + \left(66 a^{2} + 79 a + 55\right)\cdot 103^{8} + \left(9 a^{2} + 90 a + 57\right)\cdot 103^{9} +O(103^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 75 a^{2} + 100 a + 85 + \left(50 a^{2} + 19 a + 18\right)\cdot 103 + \left(28 a + 63\right)\cdot 103^{2} + \left(51 a^{2} + 56 a + 27\right)\cdot 103^{3} + \left(83 a^{2} + 29 a + 64\right)\cdot 103^{4} + \left(6 a^{2} + 71 a + 25\right)\cdot 103^{5} + \left(89 a^{2} + 18 a + 88\right)\cdot 103^{6} + \left(8 a^{2} + 85 a + 12\right)\cdot 103^{7} + \left(101 a^{2} + 29 a + 67\right)\cdot 103^{8} + \left(102 a^{2} + 22 a + 44\right)\cdot 103^{9} +O(103^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 86 a^{2} + a + 54 + \left(29 a^{2} + 90 a + 24\right)\cdot 103 + \left(35 a^{2} + 2 a + 52\right)\cdot 103^{2} + \left(59 a^{2} + 48 a + 49\right)\cdot 103^{3} + \left(85 a^{2} + 88 a + 88\right)\cdot 103^{4} + \left(67 a^{2} + 84 a + 55\right)\cdot 103^{5} + \left(6 a^{2} + 30 a + 51\right)\cdot 103^{6} + \left(29 a^{2} + 28 a + 64\right)\cdot 103^{7} + \left(87 a^{2} + 5 a + 48\right)\cdot 103^{8} + \left(2 a^{2} + 96 a + 73\right)\cdot 103^{9} +O(103^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.