Properties

Label 6.223075008.9t31.a.a
Dimension $6$
Group $S_3\wr S_3$
Conductor $223075008$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $6$
Group: $S_3\wr S_3$
Conductor: \(223075008\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 13 \cdot 31^{3} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 9.1.6915325248.1
Galois orbit size: $1$
Smallest permutation container: $S_3\wr S_3$
Parity: odd
Determinant: 1.403.2t1.a.a
Projective image: $S_3\wr S_3$
Projective stem field: Galois closure of 9.1.6915325248.1

Defining polynomial

$f(x)$$=$ \( x^{9} - x^{8} + 3x^{7} + 2x^{6} + 8x^{5} - 3x^{4} + 15x^{3} + 13x^{2} - 1 \) 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 $\mathbb{Q}_{ 101 }$: \( x^{3} + 3x + 99 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 80 a^{2} + 68 a + 10 + \left(87 a^{2} + 40 a + 52\right)\cdot 101 + \left(36 a^{2} + 16 a + 29\right)\cdot 101^{2} + \left(26 a^{2} + 92 a + 48\right)\cdot 101^{3} + \left(59 a^{2} + 70 a + 77\right)\cdot 101^{4} + \left(5 a^{2} + 16 a + 36\right)\cdot 101^{5} + \left(17 a^{2} + 66 a + 25\right)\cdot 101^{6} + \left(18 a^{2} + 40\right)\cdot 101^{7} + \left(100 a^{2} + 20 a + 21\right)\cdot 101^{8} + \left(21 a^{2} + 79 a + 51\right)\cdot 101^{9} +O(101^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 50 a^{2} + 28 a + 86 + \left(63 a^{2} + 80 a + 36\right)\cdot 101 + \left(93 a^{2} + a + 93\right)\cdot 101^{2} + \left(52 a^{2} + 94 a + 12\right)\cdot 101^{3} + \left(93 a^{2} + 45 a + 88\right)\cdot 101^{4} + \left(57 a^{2} + 55 a + 18\right)\cdot 101^{5} + \left(38 a^{2} + 33\right)\cdot 101^{6} + \left(61 a^{2} + 69 a + 38\right)\cdot 101^{7} + \left(16 a^{2} + 84 a + 30\right)\cdot 101^{8} + \left(56 a^{2} + 32 a + 21\right)\cdot 101^{9} +O(101^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 19 a^{2} + 80 a + 90 + \left(65 a^{2} + 54 a + 6\right)\cdot 101 + \left(33 a^{2} + 28 a + 23\right)\cdot 101^{2} + \left(87 a^{2} + 73 a + 69\right)\cdot 101^{3} + \left(72 a^{2} + 38 a + 3\right)\cdot 101^{4} + \left(55 a^{2} + 30 a + 36\right)\cdot 101^{5} + \left(8 a^{2} + 97 a + 8\right)\cdot 101^{6} + \left(22 a^{2} + 59 a + 48\right)\cdot 101^{7} + \left(79 a^{2} + 29 a + 80\right)\cdot 101^{8} + \left(73 a^{2} + 10 a + 53\right)\cdot 101^{9} +O(101^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 100 a^{2} + 57 a + 85 + \left(73 a^{2} + 98 a + 57\right)\cdot 101 + \left(92 a^{2} + 5 a + 91\right)\cdot 101^{2} + \left(17 a^{2} + 74 a + 43\right)\cdot 101^{3} + \left(8 a^{2} + 71 a + 18\right)\cdot 101^{4} + \left(66 a^{2} + 50 a + 35\right)\cdot 101^{5} + \left(89 a^{2} + 16 a + 34\right)\cdot 101^{6} + \left(76 a^{2} + 84 a + 69\right)\cdot 101^{7} + \left(8 a^{2} + 15 a + 14\right)\cdot 101^{8} + \left(34 a^{2} + 39 a + 78\right)\cdot 101^{9} +O(101^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 52 a^{2} + 16 a + 90 + \left(64 a^{2} + 23 a + 38\right)\cdot 101 + \left(15 a^{2} + 93 a + 38\right)\cdot 101^{2} + \left(30 a^{2} + 33 a + 68\right)\cdot 101^{3} + \left(100 a^{2} + 84 a\right)\cdot 101^{4} + \left(77 a^{2} + 95 a + 59\right)\cdot 101^{5} + \left(73 a^{2} + 83 a + 2\right)\cdot 101^{6} + \left(63 a^{2} + 48 a + 43\right)\cdot 101^{7} + \left(75 a^{2} + 47\right)\cdot 101^{8} + \left(10 a^{2} + 29 a + 31\right)\cdot 101^{9} +O(101^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 2 a^{2} + 54 a + 56 + \left(49 a^{2} + 5 a + 75\right)\cdot 101 + \left(30 a^{2} + 56 a + 16\right)\cdot 101^{2} + \left(88 a^{2} + 36 a + 71\right)\cdot 101^{3} + \left(69 a^{2} + 92 a + 98\right)\cdot 101^{4} + \left(39 a^{2} + 53 a + 3\right)\cdot 101^{5} + \left(75 a^{2} + 38 a + 41\right)\cdot 101^{6} + \left(60 a^{2} + 40 a + 24\right)\cdot 101^{7} + \left(22 a^{2} + 51 a + 68\right)\cdot 101^{8} + \left(5 a^{2} + 11 a + 17\right)\cdot 101^{9} +O(101^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 54 a^{2} + 52 a + 3 + \left(77 a^{2} + 5 a + 32\right)\cdot 101 + \left(68 a^{2} + 81 a + 6\right)\cdot 101^{2} + \left(19 a^{2} + 94 a + 2\right)\cdot 101^{3} + \left(79 a^{2} + 52 a + 29\right)\cdot 101^{4} + \left(85 a^{2} + 26 a + 7\right)\cdot 101^{5} + \left(80 a^{2} + 89 a + 46\right)\cdot 101^{6} + \left(43 a^{2} + 59 a + 33\right)\cdot 101^{7} + \left(9 a^{2} + 63 a + 32\right)\cdot 101^{8} + \left(48 a^{2} + 10 a + 45\right)\cdot 101^{9} +O(101^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 81 a^{2} + 7 a + 57 + \left(62 a^{2} + 89 a + 2\right)\cdot 101 + \left(100 a^{2} + 22 a + 70\right)\cdot 101^{2} + \left(75 a^{2} + 48 a + 13\right)\cdot 101^{3} + \left(93 a^{2} + 24 a + 58\right)\cdot 101^{4} + \left(26 a^{2} + 48 a + 91\right)\cdot 101^{5} + \left(8 a^{2} + 54 a + 1\right)\cdot 101^{6} + \left(54 a^{2} + 90 a + 54\right)\cdot 101^{7} + \left(62 a^{2} + 16 a + 37\right)\cdot 101^{8} + \left(59 a^{2} + 83 a + 68\right)\cdot 101^{9} +O(101^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 67 a^{2} + 42 a + 29 + \left(61 a^{2} + 6 a\right)\cdot 101 + \left(32 a^{2} + 98 a + 35\right)\cdot 101^{2} + \left(5 a^{2} + 58 a + 74\right)\cdot 101^{3} + \left(29 a^{2} + 23 a + 29\right)\cdot 101^{4} + \left(89 a^{2} + 26 a + 14\right)\cdot 101^{5} + \left(11 a^{2} + 58 a + 9\right)\cdot 101^{6} + \left(3 a^{2} + 51 a + 53\right)\cdot 101^{7} + \left(29 a^{2} + 20 a + 71\right)\cdot 101^{8} + \left(94 a^{2} + 7 a + 36\right)\cdot 101^{9} +O(101^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.