Properties

Label 12.239...784.36t2210.a.a
Dimension $12$
Group $S_3\wr S_3$
Conductor $2.399\times 10^{21}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $12$
Group: $S_3\wr S_3$
Conductor: \(239\!\cdots\!784\)\(\medspace = 2^{20} \cdot 3^{6} \cdot 11^{12} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 9.1.164627620608.1
Galois orbit size: $1$
Smallest permutation container: 36T2210
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $S_3\wr S_3$
Projective stem field: Galois closure of 9.1.164627620608.1

Defining polynomial

$f(x)$$=$ \( x^{9} - x^{8} - 2x^{7} + 5x^{6} - 10x^{5} + 6x^{4} + 24x^{3} - 20x^{2} - 8x + 24 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 97 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$: \( x^{3} + 9x + 92 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 91 a^{2} + 87 a + 56 + \left(87 a^{2} + 55 a + 75\right)\cdot 97 + \left(45 a^{2} + 53 a + 64\right)\cdot 97^{2} + \left(17 a^{2} + 61 a + 28\right)\cdot 97^{3} + \left(82 a^{2} + 16 a + 90\right)\cdot 97^{4} + \left(76 a^{2} + 9 a + 4\right)\cdot 97^{5} + \left(55 a^{2} + 43 a + 58\right)\cdot 97^{6} + \left(12 a^{2} + 65 a + 88\right)\cdot 97^{7} + \left(39 a^{2} + 84 a + 72\right)\cdot 97^{8} + \left(95 a^{2} + 4 a + 25\right)\cdot 97^{9} +O(97^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 44 a^{2} + 87 a + 72 + \left(56 a^{2} + 94 a + 37\right)\cdot 97 + \left(39 a^{2} + 89 a + 27\right)\cdot 97^{2} + \left(87 a^{2} + 39 a + 9\right)\cdot 97^{3} + \left(89 a^{2} + 40 a + 49\right)\cdot 97^{4} + \left(35 a^{2} + 79 a + 56\right)\cdot 97^{5} + \left(32 a^{2} + 42 a + 61\right)\cdot 97^{6} + \left(56 a^{2} + 76 a + 20\right)\cdot 97^{7} + \left(65 a^{2} + 83 a + 67\right)\cdot 97^{8} + \left(86 a^{2} + 95 a + 68\right)\cdot 97^{9} +O(97^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 18 a^{2} + 93 a + 13 + \left(94 a^{2} + 51 a + 70\right)\cdot 97 + \left(78 a^{2} + 90 a + 69\right)\cdot 97^{2} + \left(37 a^{2} + 76 a + 2\right)\cdot 97^{3} + \left(29 a^{2} + 59 a + 74\right)\cdot 97^{4} + \left(82 a^{2} + 65 a + 43\right)\cdot 97^{5} + \left(78 a^{2} + 83 a + 49\right)\cdot 97^{6} + \left(86 a^{2} + 22 a + 9\right)\cdot 97^{7} + \left(34 a^{2} + 7 a + 77\right)\cdot 97^{8} + \left(81 a^{2} + 63 a + 36\right)\cdot 97^{9} +O(97^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 69 a^{2} + 43 a + 94 + \left(13 a^{2} + 17 a + 26\right)\cdot 97 + \left(22 a^{2} + 85 a + 36\right)\cdot 97^{2} + \left(45 a^{2} + 48 a + 54\right)\cdot 97^{3} + \left(81 a^{2} + 70 a + 88\right)\cdot 97^{4} + \left(12 a^{2} + 32 a + 77\right)\cdot 97^{5} + \left(39 a^{2} + 22 a + 29\right)\cdot 97^{6} + \left(86 a^{2} + 77 a + 14\right)\cdot 97^{7} + \left(44 a^{2} + 84 a + 46\right)\cdot 97^{8} + \left(5 a^{2} + 81 a + 28\right)\cdot 97^{9} +O(97^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 35 a^{2} + 83 a + 11 + \left(49 a^{2} + 79 a + 38\right)\cdot 97 + \left(47 a^{2} + 72 a + 74\right)\cdot 97^{2} + \left(18 a^{2} + 23 a + 34\right)\cdot 97^{3} + \left(27 a^{2} + 64 a + 51\right)\cdot 97^{4} + \left(11 a^{2} + 8 a + 96\right)\cdot 97^{5} + \left(55 a^{2} + 60 a + 53\right)\cdot 97^{6} + \left(78 a^{2} + 53 a + 96\right)\cdot 97^{7} + \left(48 a^{2} + 77 a + 33\right)\cdot 97^{8} + \left(25 a^{2} + 77 a + 91\right)\cdot 97^{9} +O(97^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 68 a^{2} + 96 a + 88 + \left(7 a^{2} + 35 a + 87\right)\cdot 97 + \left(72 a^{2} + 52 a + 44\right)\cdot 97^{2} + \left(36 a^{2} + 47 a + 3\right)\cdot 97^{3} + \left(23 a^{2} + 9 a + 31\right)\cdot 97^{4} + \left(69 a^{2} + 30 a + 28\right)\cdot 97^{5} + \left(26 a^{2} + 68 a + 52\right)\cdot 97^{6} + \left(34 a^{2} + 77 a + 89\right)\cdot 97^{7} + \left(45 a^{2} + 70 a + 48\right)\cdot 97^{8} + \left(44 a^{2} + 82 a + 68\right)\cdot 97^{9} +O(97^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 35 a^{2} + 14 a + 18 + \left(43 a^{2} + 47 a + 56\right)\cdot 97 + \left(75 a^{2} + 13 a + 48\right)\cdot 97^{2} + \left(68 a^{2} + 77 a + 91\right)\cdot 97^{3} + \left(74 a^{2} + 93 a + 54\right)\cdot 97^{4} + \left(75 a^{2} + 48 a + 4\right)\cdot 97^{5} + \left(82 a^{2} + 67 a + 73\right)\cdot 97^{6} + \left(50 a^{2} + 94 a + 84\right)\cdot 97^{7} + \left(93 a^{2} + 5 a + 40\right)\cdot 97^{8} + \left(25 a^{2} + 35 a + 92\right)\cdot 97^{9} +O(97^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 68 a^{2} + 24 a + 15 + \left(56 a^{2} + 58 a + 82\right)\cdot 97 + \left(3 a^{2} + 67 a + 4\right)\cdot 97^{2} + \left(61 a^{2} + 11 a + 96\right)\cdot 97^{3} + \left(84 a^{2} + 16 a + 7\right)\cdot 97^{4} + \left(8 a^{2} + 79 a + 82\right)\cdot 97^{5} + \left(83 a^{2} + 90 a + 27\right)\cdot 97^{6} + \left(5 a^{2} + 74 a + 48\right)\cdot 97^{7} + \left(9 a^{2} + 31 a + 86\right)\cdot 97^{8} + \left(73 a^{2} + 14 a + 85\right)\cdot 97^{9} +O(97^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 57 a^{2} + 55 a + 22 + \left(75 a^{2} + 43 a + 10\right)\cdot 97 + \left(2 a^{2} + 56 a + 17\right)\cdot 97^{2} + \left(15 a^{2} + 67\right)\cdot 97^{3} + \left(89 a^{2} + 17 a + 37\right)\cdot 97^{4} + \left(14 a^{2} + 34 a + 90\right)\cdot 97^{5} + \left(31 a^{2} + 6 a + 78\right)\cdot 97^{6} + \left(73 a^{2} + 39 a + 32\right)\cdot 97^{7} + \left(6 a^{2} + 38 a + 11\right)\cdot 97^{8} + \left(47 a^{2} + 29 a + 84\right)\cdot 97^{9} +O(97^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.