Properties

Label 6.3102044416.9t30.a.a
Dimension $6$
Group $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$
Conductor $3102044416$
Root number $1$
Indicator $1$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $6$
Group: $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$
Conductor: \(3102044416\)\(\medspace = 2^{8} \cdot 59^{4} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 9.1.43192866448384.1
Galois orbit size: $1$
Smallest permutation container: $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$
Parity: odd
Determinant: 1.4.2t1.a.a
Projective image: $C_3^3:S_4$
Projective stem field: Galois closure of 9.1.43192866448384.1

Defining polynomial

$f(x)$$=$ \( x^{9} - 3x^{8} - 7x^{7} + 21x^{6} + 39x^{5} - 25x^{4} - 77x^{3} - 25x^{2} + 48x + 44 \) 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 $\mathbb{Q}_{ 101 }$: \( x^{3} + 3x + 99 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 98 a^{2} + 26 a + 66 + \left(17 a^{2} + 63 a + 95\right)\cdot 101 + \left(83 a^{2} + 98 a + 38\right)\cdot 101^{2} + \left(79 a^{2} + 85 a + 15\right)\cdot 101^{3} + \left(27 a^{2} + 44 a + 56\right)\cdot 101^{4} + \left(98 a^{2} + 59 a + 44\right)\cdot 101^{5} + \left(18 a^{2} + 58 a + 73\right)\cdot 101^{6} + \left(40 a^{2} + 17 a + 94\right)\cdot 101^{7} + \left(78 a^{2} + 5 a + 94\right)\cdot 101^{8} + \left(a^{2} + 83 a + 31\right)\cdot 101^{9} +O(101^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 56 a^{2} + 11 a + 83 + \left(71 a^{2} + 30 a + 76\right)\cdot 101 + \left(58 a^{2} + 71 a + 57\right)\cdot 101^{2} + \left(44 a^{2} + 27 a + 11\right)\cdot 101^{3} + \left(19 a^{2} + 14 a + 27\right)\cdot 101^{4} + \left(69 a^{2} + 51 a + 51\right)\cdot 101^{5} + \left(76 a^{2} + 78 a + 77\right)\cdot 101^{6} + \left(43 a^{2} + 26 a + 96\right)\cdot 101^{7} + \left(37 a^{2} + 71 a + 7\right)\cdot 101^{8} + \left(3 a^{2} + 37 a + 83\right)\cdot 101^{9} +O(101^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 79 a^{2} + 63 a + 28 + \left(13 a^{2} + 34 a + 62\right)\cdot 101 + \left(13 a^{2} + 80 a + 67\right)\cdot 101^{2} + \left(70 a^{2} + 55 a + 62\right)\cdot 101^{3} + \left(31 a^{2} + 58 a + 51\right)\cdot 101^{4} + \left(5 a^{2} + 45 a + 24\right)\cdot 101^{5} + \left(60 a^{2} + 91 a + 44\right)\cdot 101^{6} + \left(83 a^{2} + 99 a + 75\right)\cdot 101^{7} + \left(79 a^{2} + 2 a + 92\right)\cdot 101^{8} + \left(22 a^{2} + 9 a + 20\right)\cdot 101^{9} +O(101^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 20 a^{2} + 21 a + 99 + \left(73 a^{2} + 99 a + 51\right)\cdot 101 + \left(44 a^{2} + 78 a + 74\right)\cdot 101^{2} + \left(33 a^{2} + 10 a + 86\right)\cdot 101^{3} + \left(41 a^{2} + 63 a + 93\right)\cdot 101^{4} + \left(10 a^{2} + 67 a + 57\right)\cdot 101^{5} + \left(48 a^{2} + 37 a + 35\right)\cdot 101^{6} + \left(41 a^{2} + 95 a + 59\right)\cdot 101^{7} + \left(11 a^{2} + 55 a + 50\right)\cdot 101^{8} + \left(44 a^{2} + 71 a + 84\right)\cdot 101^{9} +O(101^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 2 a^{2} + 17 a + 76 + \left(14 a^{2} + 68 a + 87\right)\cdot 101 + \left(43 a^{2} + 42 a + 59\right)\cdot 101^{2} + \left(98 a^{2} + 34 a + 52\right)\cdot 101^{3} + \left(27 a^{2} + 80 a + 56\right)\cdot 101^{4} + \left(85 a^{2} + 88 a + 18\right)\cdot 101^{5} + \left(93 a^{2} + 72 a + 21\right)\cdot 101^{6} + \left(76 a^{2} + 6 a + 67\right)\cdot 101^{7} + \left(9 a^{2} + 42 a + 58\right)\cdot 101^{8} + \left(34 a^{2} + 20 a + 96\right)\cdot 101^{9} +O(101^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 67 a^{2} + 27 a + 4 + \left(15 a^{2} + 36 a + 66\right)\cdot 101 + \left(29 a^{2} + 50 a + 99\right)\cdot 101^{2} + \left(87 a^{2} + 17 a + 96\right)\cdot 101^{3} + \left(49 a^{2} + 28 a + 87\right)\cdot 101^{4} + \left(26 a^{2} + 4 a + 66\right)\cdot 101^{5} + \left(65 a^{2} + 32 a + 54\right)\cdot 101^{6} + \left(74 a^{2} + 75 a + 57\right)\cdot 101^{7} + \left(84 a^{2} + 26 a + 1\right)\cdot 101^{8} + \left(74 a^{2} + 54 a + 24\right)\cdot 101^{9} +O(101^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( a^{2} + 58 a + 74 + \left(69 a^{2} + 70 a + 96\right)\cdot 101 + \left(75 a^{2} + 60 a + 23\right)\cdot 101^{2} + \left(23 a^{2} + 81 a + 4\right)\cdot 101^{3} + \left(45 a^{2} + 76 a + 91\right)\cdot 101^{4} + \left(18 a^{2} + 53 a + 86\right)\cdot 101^{5} + \left(89 a^{2} + 70 a + 11\right)\cdot 101^{6} + \left(84 a^{2} + 76 a + 83\right)\cdot 101^{7} + \left(12 a^{2} + 53 a + 64\right)\cdot 101^{8} + \left(65 a^{2} + 98 a + 57\right)\cdot 101^{9} +O(101^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 44 a^{2} + 32 a + 46 + \left(61 a^{2} + 28\right)\cdot 101 + \left(67 a^{2} + 70 a + 19\right)\cdot 101^{2} + \left(32 a^{2} + 92 a + 85\right)\cdot 101^{3} + \left(36 a^{2} + 9 a + 83\right)\cdot 101^{4} + \left(13 a^{2} + 97 a + 63\right)\cdot 101^{5} + \left(36 a^{2} + 52 a + 11\right)\cdot 101^{6} + \left(73 a^{2} + 98 a + 22\right)\cdot 101^{7} + \left(50 a^{2} + 76 a + 28\right)\cdot 101^{8} + \left(32 a^{2} + 65 a + 61\right)\cdot 101^{9} +O(101^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 37 a^{2} + 48 a + 32 + \left(67 a^{2} + a + 40\right)\cdot 101 + \left(89 a^{2} + 53 a + 63\right)\cdot 101^{2} + \left(34 a^{2} + 98 a + 89\right)\cdot 101^{3} + \left(23 a^{2} + 27 a + 57\right)\cdot 101^{4} + \left(77 a^{2} + 37 a + 90\right)\cdot 101^{5} + \left(16 a^{2} + 10 a + 73\right)\cdot 101^{6} + \left(87 a^{2} + 8 a + 49\right)\cdot 101^{7} + \left(38 a^{2} + 69 a + 4\right)\cdot 101^{8} + \left(24 a^{2} + 64 a + 45\right)\cdot 101^{9} +O(101^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.