Properties

Label 9.828465164288.10t32.a.a
Dimension $9$
Group $S_6$
Conductor $828465164288$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $9$
Group: $S_6$
Conductor: \(828465164288\)\(\medspace = 2^{12} \cdot 587^{3}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 6.2.37568.1
Galois orbit size: $1$
Smallest permutation container: $S_{6}$
Parity: even
Determinant: 1.2348.2t1.a.a
Projective image: $S_6$
Projective field: Galois closure of 6.2.37568.1

Defining polynomial

$f(x)$$=$$ x^{6} - 2 x^{5} + 2 x^{4} - x^{2} + 2 x - 1 $.

The roots of $f$ are computed in an extension of $\Q_{ 167 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 167 }$: $ x^{2} + 166 x + 5 $

Roots:
$r_{ 1 }$ $=$ $ 21 a + 3 + \left(150 a + 82\right)\cdot 167 + \left(89 a + 154\right)\cdot 167^{2} + \left(128 a + 73\right)\cdot 167^{3} + \left(142 a + 128\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 85 a + 8 + \left(67 a + 69\right)\cdot 167 + \left(33 a + 122\right)\cdot 167^{2} + \left(97 a + 61\right)\cdot 167^{3} + \left(128 a + 31\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 146 a + 24 + \left(16 a + 44\right)\cdot 167 + \left(77 a + 94\right)\cdot 167^{2} + \left(38 a + 112\right)\cdot 167^{3} + \left(24 a + 142\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 31 a + 5 + \left(60 a + 29\right)\cdot 167 + \left(104 a + 82\right)\cdot 167^{2} + \left(106 a + 62\right)\cdot 167^{3} + \left(58 a + 8\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 82 a + 93 + \left(99 a + 51\right)\cdot 167 + \left(133 a + 88\right)\cdot 167^{2} + \left(69 a + 125\right)\cdot 167^{3} + \left(38 a + 62\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 136 a + 36 + \left(106 a + 58\right)\cdot 167 + \left(62 a + 126\right)\cdot 167^{2} + \left(60 a + 64\right)\cdot 167^{3} + \left(108 a + 127\right)\cdot 167^{4} +O\left(167^{ 5 }\right)$

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.