Properties

Label 12.285...408.36t2216.a.a
Dimension $12$
Group $S_3\wr S_3$
Conductor $2.856\times 10^{19}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $12$
Group: $S_3\wr S_3$
Conductor: \(28563402308314025408\)\(\medspace = 2^{6} \cdot 23^{7} \cdot 107^{4} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 9.1.83319616.1
Galois orbit size: $1$
Smallest permutation container: 36T2216
Parity: odd
Determinant: 1.23.2t1.a.a
Projective image: $S_3\wr S_3$
Projective stem field: Galois closure of 9.1.83319616.1

Defining polynomial

$f(x)$$=$ \( x^{9} - 4x^{8} + 9x^{7} - 15x^{6} + 17x^{5} - 14x^{4} + 8x^{3} - x^{2} - x + 1 \) 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:
$r_{ 1 }$ $=$ \( 45 a^{2} + 93 a + 92 + \left(99 a^{2} + 54 a + 25\right)\cdot 101 + \left(2 a^{2} + 78 a + 2\right)\cdot 101^{2} + \left(3 a^{2} + 44 a + 2\right)\cdot 101^{3} + \left(64 a^{2} + 96 a + 47\right)\cdot 101^{4} + \left(66 a^{2} + 56 a + 2\right)\cdot 101^{5} + \left(67 a^{2} + 13 a + 62\right)\cdot 101^{6} + \left(63 a^{2} + 50 a + 60\right)\cdot 101^{7} + \left(a^{2} + 82 a + 59\right)\cdot 101^{8} + \left(63 a^{2} + 33 a + 71\right)\cdot 101^{9} +O(101^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 42 a^{2} + 68 a + 97 + \left(8 a^{2} + 92 a + 1\right)\cdot 101 + \left(74 a^{2} + 47 a + 93\right)\cdot 101^{2} + \left(36 a^{2} + 29 a + 13\right)\cdot 101^{3} + \left(72 a^{2} + 94 a + 38\right)\cdot 101^{4} + \left(88 a^{2} + 16 a + 39\right)\cdot 101^{5} + \left(38 a^{2} + 14 a + 71\right)\cdot 101^{6} + \left(86 a^{2} + 69 a + 54\right)\cdot 101^{7} + \left(90 a^{2} + 83 a + 27\right)\cdot 101^{8} + \left(25 a^{2} + 47 a + 78\right)\cdot 101^{9} +O(101^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 90 a^{2} + 86 a + 92 + \left(a^{2} + 71 a + 89\right)\cdot 101 + \left(20 a^{2} + 22 a + 85\right)\cdot 101^{2} + \left(36 a^{2} + 34 a + 12\right)\cdot 101^{3} + \left(58 a^{2} + 50 a + 10\right)\cdot 101^{4} + \left(26 a^{2} + 19 a + 16\right)\cdot 101^{5} + \left(37 a^{2} + 96 a + 68\right)\cdot 101^{6} + \left(88 a^{2} + 45 a + 58\right)\cdot 101^{7} + \left(3 a^{2} + 18 a + 55\right)\cdot 101^{8} + \left(99 a^{2} + 40 a + 22\right)\cdot 101^{9} +O(101^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 57 + 52\cdot 101 + 18\cdot 101^{2} + 15\cdot 101^{3} + 43\cdot 101^{4} + 46\cdot 101^{5} + 37\cdot 101^{6} + 36\cdot 101^{7} + 9\cdot 101^{8} + 16\cdot 101^{9} +O(101^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 42 + 90\cdot 101 + 99\cdot 101^{2} + 84\cdot 101^{3} + 86\cdot 101^{4} + 63\cdot 101^{5} + 53\cdot 101^{6} + 60\cdot 101^{7} + 69\cdot 101^{8} + 81\cdot 101^{9} +O(101^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 70 a^{2} + 48 a + 52 + \left(90 a^{2} + 37 a + 65\right)\cdot 101 + \left(6 a^{2} + 30 a + 59\right)\cdot 101^{2} + \left(28 a^{2} + 37 a + 97\right)\cdot 101^{3} + \left(71 a^{2} + 57 a + 35\right)\cdot 101^{4} + \left(86 a^{2} + 64 a + 35\right)\cdot 101^{5} + \left(24 a^{2} + 91 a + 43\right)\cdot 101^{6} + \left(27 a^{2} + 86 a + 37\right)\cdot 101^{7} + \left(6 a^{2} + 99 a + 60\right)\cdot 101^{8} + \left(77 a^{2} + 12 a + 79\right)\cdot 101^{9} +O(101^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 62 + 16\cdot 101 + 58\cdot 101^{2} + 90\cdot 101^{3} + 28\cdot 101^{4} + 90\cdot 101^{5} + 46\cdot 101^{6} + 53\cdot 101^{7} + 12\cdot 101^{8} + 87\cdot 101^{9} +O(101^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 62 a^{2} + 26 a + 25 + \left(83 a^{2} + 50 a + 95\right)\cdot 101 + \left(96 a^{2} + 16 a + 88\right)\cdot 101^{2} + \left(50 a^{2} + 80 a + 97\right)\cdot 101^{3} + \left(91 a^{2} + 97 a\right)\cdot 101^{4} + \left(63 a^{2} + 87 a + 98\right)\cdot 101^{5} + \left(25 a^{2} + 75 a + 78\right)\cdot 101^{6} + \left(94 a^{2} + 9 a + 20\right)\cdot 101^{7} + \left(70 a^{2} + 15 a + 97\right)\cdot 101^{8} + \left(40 a^{2} + 8 a + 26\right)\cdot 101^{9} +O(101^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 95 a^{2} + 83 a + 91 + \left(18 a^{2} + 96 a + 66\right)\cdot 101 + \left(a^{2} + 5 a + 99\right)\cdot 101^{2} + \left(47 a^{2} + 77 a + 89\right)\cdot 101^{3} + \left(46 a^{2} + 7 a + 11\right)\cdot 101^{4} + \left(71 a^{2} + 57 a + 12\right)\cdot 101^{5} + \left(7 a^{2} + 11 a + 43\right)\cdot 101^{6} + \left(44 a^{2} + 41 a + 21\right)\cdot 101^{7} + \left(28 a^{2} + 3 a + 12\right)\cdot 101^{8} + \left(98 a^{2} + 59 a + 41\right)\cdot 101^{9} +O(101^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.