Properties

Label 8.781...632.18t157.a.a
Dimension $8$
Group $((C_3^2:Q_8):C_3):C_2$
Conductor $7.816\times 10^{14}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $8$
Group: $((C_3^2:Q_8):C_3):C_2$
Conductor: \(781589632957632\)\(\medspace = 2^{6} \cdot 3^{7} \cdot 89^{5} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 9.3.98673100992.1
Galois orbit size: $1$
Smallest permutation container: 18T157
Parity: odd
Determinant: 1.267.2t1.a.a
Projective image: $C_3^2:\GL(2,3)$
Projective stem field: Galois closure of 9.3.98673100992.1

Defining polynomial

$f(x)$$=$ \( x^{9} - 3x^{8} + 3x^{7} - 6x^{6} + 12x^{5} + 3x^{4} - 21x^{3} + 9x^{2} + 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 101 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 101 }$: \( x^{4} + x^{2} + 78x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 81 + 42\cdot 101 + 48\cdot 101^{2} + 91\cdot 101^{3} + 20\cdot 101^{4} + 63\cdot 101^{5} + 96\cdot 101^{6} + 55\cdot 101^{7} + 95\cdot 101^{8} + 55\cdot 101^{9} +O(101^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 64 a^{3} + 4 a^{2} + 32 a + 6 + \left(86 a^{3} + a^{2} + 67 a + 40\right)\cdot 101 + \left(78 a^{3} + 36 a^{2} + 7 a + 69\right)\cdot 101^{2} + \left(14 a^{3} + 10 a^{2} + 99 a + 19\right)\cdot 101^{3} + \left(68 a^{3} + 48 a^{2} + 32 a + 84\right)\cdot 101^{4} + \left(7 a^{3} + 39 a^{2} + 31 a + 27\right)\cdot 101^{5} + \left(67 a^{3} + 17 a^{2} + 92 a + 25\right)\cdot 101^{6} + \left(97 a^{3} + 16 a^{2} + 8 a + 65\right)\cdot 101^{7} + \left(a^{3} + 55 a^{2} + 39 a + 77\right)\cdot 101^{8} + \left(33 a^{3} + 79 a^{2} + 73 a + 76\right)\cdot 101^{9} +O(101^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 99 a^{3} + 80 a^{2} + 15 a + 58 + \left(81 a^{3} + 99 a^{2} + 62 a + 66\right)\cdot 101 + \left(54 a^{3} + 70 a^{2} + 79 a + 14\right)\cdot 101^{2} + \left(67 a^{3} + 16 a^{2} + 18 a + 65\right)\cdot 101^{3} + \left(49 a^{3} + 69 a^{2} + 4 a + 18\right)\cdot 101^{4} + \left(90 a^{3} + 34 a^{2} + 12 a + 80\right)\cdot 101^{5} + \left(91 a^{3} + 4 a^{2} + 64 a + 19\right)\cdot 101^{6} + \left(82 a^{3} + 54 a^{2} + 87 a + 91\right)\cdot 101^{7} + \left(100 a^{3} + 43 a^{2} + 70 a + 1\right)\cdot 101^{8} + \left(31 a^{3} + 59 a^{2} + 90 a + 20\right)\cdot 101^{9} +O(101^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 61 a^{3} + 4 a^{2} + 46 a + 83 + \left(4 a^{2} + 67 a + 58\right)\cdot 101 + \left(8 a^{3} + 16 a^{2} + 78 a + 55\right)\cdot 101^{2} + \left(82 a^{3} + 28 a^{2} + 57 a + 26\right)\cdot 101^{3} + \left(20 a^{3} + 72 a^{2} + 41 a + 54\right)\cdot 101^{4} + \left(14 a^{3} + 91 a^{2} + 79 a + 82\right)\cdot 101^{5} + \left(43 a^{3} + 72 a^{2} + 72 a + 66\right)\cdot 101^{6} + \left(80 a^{3} + 78 a^{2} + 19 a + 47\right)\cdot 101^{7} + \left(10 a^{3} + 46 a^{2} + 9 a + 34\right)\cdot 101^{8} + \left(2 a^{3} + 18 a^{2} + 22 a + 5\right)\cdot 101^{9} +O(101^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 30 a^{3} + 44 a^{2} + 61 a + 94 + \left(98 a^{3} + 70 a^{2} + 62 a + 46\right)\cdot 101 + \left(21 a^{3} + 47 a^{2} + 66 a + 51\right)\cdot 101^{2} + \left(48 a^{3} + 10 a^{2} + 87 a + 42\right)\cdot 101^{3} + \left(62 a^{3} + 62 a^{2} + 9 a + 57\right)\cdot 101^{4} + \left(96 a^{3} + 27 a^{2} + 41 a + 81\right)\cdot 101^{5} + \left(82 a^{3} + 73 a^{2} + 100 a + 86\right)\cdot 101^{6} + \left(54 a^{3} + 49 a^{2} + 21 a + 61\right)\cdot 101^{7} + \left(38 a^{3} + 24 a^{2} + 24 a + 35\right)\cdot 101^{8} + \left(99 a^{3} + 62 a^{2} + 96 a + 24\right)\cdot 101^{9} +O(101^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 35 a^{3} + 71 a^{2} + 71 a + 60 + \left(69 a^{3} + 98 a^{2} + 43 a + 87\right)\cdot 101 + \left(65 a^{3} + 65 a^{2} + 51 a + 20\right)\cdot 101^{2} + \left(74 a^{3} + 13 a^{2} + 31 a + 39\right)\cdot 101^{3} + \left(75 a^{3} + 93 a^{2} + 16 a + 96\right)\cdot 101^{4} + \left(67 a^{3} + 26 a^{2} + 89 a\right)\cdot 101^{5} + \left(15 a^{3} + 7 a^{2} + 50 a + 43\right)\cdot 101^{6} + \left(22 a^{3} + 64 a^{2} + 32 a + 65\right)\cdot 101^{7} + \left(86 a^{3} + 42 a^{2} + 82 a\right)\cdot 101^{8} + \left(57 a^{3} + 44 a^{2} + 100 a + 98\right)\cdot 101^{9} +O(101^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 4 a^{3} + 47 a^{2} + 84 a + 39 + \left(19 a^{3} + 10 a^{2} + 74 a + 26\right)\cdot 101 + \left(25 a^{3} + 14 a^{2} + 20 a + 71\right)\cdot 101^{2} + \left(74 a^{3} + 34 a^{2} + 26 a + 11\right)\cdot 101^{3} + \left(27 a^{3} + 63 a^{2} + 37 a + 96\right)\cdot 101^{4} + \left(23 a^{3} + 79 a^{2} + 65 a + 58\right)\cdot 101^{5} + \left(18 a^{3} + 29 a^{2} + 49 a + 15\right)\cdot 101^{6} + \left(21 a^{3} + 89 a^{2} + 74 a + 83\right)\cdot 101^{7} + \left(46 a^{3} + 9 a^{2} + 10 a + 72\right)\cdot 101^{8} + \left(71 a^{3} + 18 a^{2} + 60 a + 85\right)\cdot 101^{9} +O(101^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 42 a^{3} + 22 a^{2} + 53 a + 41 + \left(45 a^{3} + 98 a^{2} + 23 a\right)\cdot 101 + \left(49 a^{3} + 83 a^{2} + 64 a + 90\right)\cdot 101^{2} + \left(30 a^{3} + 48 a^{2} + 13 a + 48\right)\cdot 101^{3} + \left(37 a^{3} + 89 a^{2} + 10 a + 17\right)\cdot 101^{4} + \left(11 a^{3} + 43 a^{2} + 2 a + 44\right)\cdot 101^{5} + \left(76 a^{3} + 3 a^{2} + 87 a + 92\right)\cdot 101^{6} + \left(a^{3} + 43 a^{2} + 39 a + 22\right)\cdot 101^{7} + \left(2 a^{3} + 57 a^{2} + 71 a + 31\right)\cdot 101^{8} + \left(8 a^{3} + 59 a^{2} + 5 a + 18\right)\cdot 101^{9} +O(101^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 69 a^{3} + 31 a^{2} + 42 a + 46 + \left(2 a^{3} + 21 a^{2} + 2 a + 34\right)\cdot 101 + \left(100 a^{3} + 69 a^{2} + 35 a + 83\right)\cdot 101^{2} + \left(11 a^{3} + 39 a^{2} + 69 a + 58\right)\cdot 101^{3} + \left(62 a^{3} + 7 a^{2} + 49 a + 59\right)\cdot 101^{4} + \left(92 a^{3} + 60 a^{2} + 83 a + 65\right)\cdot 101^{5} + \left(8 a^{3} + 94 a^{2} + 88 a + 58\right)\cdot 101^{6} + \left(43 a^{3} + 8 a^{2} + 17 a + 11\right)\cdot 101^{7} + \left(16 a^{3} + 23 a^{2} + 96 a + 54\right)\cdot 101^{8} + \left(100 a^{3} + 62 a^{2} + 55 a + 19\right)\cdot 101^{9} +O(101^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 9 }$ Character value
$1$$1$$()$$8$
$9$$2$$(1,4)(2,3)(5,7)(6,9)$$0$
$36$$2$$(1,4)(5,6)(7,9)$$-2$
$8$$3$$(1,2,6)(3,4,9)(5,8,7)$$-1$
$24$$3$$(1,7,3)(2,4,5)$$2$
$48$$3$$(1,7,9)(2,5,3)(4,6,8)$$-1$
$54$$4$$(1,3,4,2)(5,6,7,9)$$0$
$72$$6$$(1,6,9,4,5,7)(2,8,3)$$1$
$72$$6$$(1,7)(2,6,4,8,5,9)$$0$
$54$$8$$(1,6,3,7,4,9,2,5)$$0$
$54$$8$$(1,9,3,5,4,6,2,7)$$0$

The blue line marks the conjugacy class containing complex conjugation.