Properties

Label 9.168...072.10t32.a.a
Dimension $9$
Group $S_6$
Conductor $1.680\times 10^{12}$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $9$
Group: $S_6$
Conductor: \(1680066179072\)\(\medspace = 2^{12} \cdot 743^{3} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.0.47552.1
Galois orbit size: $1$
Smallest permutation container: $S_{6}$
Parity: odd
Determinant: 1.743.2t1.a.a
Projective image: $S_6$
Projective stem field: Galois closure of 6.0.47552.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 131 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 131 }$: \( x^{2} + 127x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 128 + 104\cdot 131 + 67\cdot 131^{2} + 47\cdot 131^{3} + 2\cdot 131^{4} +O(131^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 18 a + 12 + \left(101 a + 56\right)\cdot 131 + \left(67 a + 25\right)\cdot 131^{2} + \left(98 a + 127\right)\cdot 131^{3} + \left(63 a + 119\right)\cdot 131^{4} +O(131^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 113 a + 84 + \left(29 a + 49\right)\cdot 131 + \left(63 a + 64\right)\cdot 131^{2} + \left(32 a + 60\right)\cdot 131^{3} + \left(67 a + 14\right)\cdot 131^{4} +O(131^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 6 a + 81 + \left(57 a + 10\right)\cdot 131 + \left(6 a + 104\right)\cdot 131^{2} + \left(64 a + 97\right)\cdot 131^{3} + \left(12 a + 51\right)\cdot 131^{4} +O(131^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 116 + 69\cdot 131 + 58\cdot 131^{2} + 105\cdot 131^{3} + 35\cdot 131^{4} +O(131^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 125 a + 105 + \left(73 a + 101\right)\cdot 131 + \left(124 a + 72\right)\cdot 131^{2} + \left(66 a + 85\right)\cdot 131^{3} + \left(118 a + 37\right)\cdot 131^{4} +O(131^{5})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$9$
$15$$2$$(1,2)(3,4)(5,6)$$3$
$15$$2$$(1,2)$$3$
$45$$2$$(1,2)(3,4)$$1$
$40$$3$$(1,2,3)(4,5,6)$$0$
$40$$3$$(1,2,3)$$0$
$90$$4$$(1,2,3,4)(5,6)$$1$
$90$$4$$(1,2,3,4)$$-1$
$144$$5$$(1,2,3,4,5)$$-1$
$120$$6$$(1,2,3,4,5,6)$$0$
$120$$6$$(1,2,3)(4,5)$$0$

The blue line marks the conjugacy class containing complex conjugation.