Galois orbit of Artin representations 9.1719926784000000.20t145.a

• Label: 9.171...000.20t145.a
• Dimension: $9$
• Group: $S_6$
• Conductor: $1.720\times 10^{15}$
• Indicator: $1$

Basic invariants

Dimension:	$9$
Group:	$S_6$
Conductor:	\(1719926784000000\)\(\medspace = 2^{24} \cdot 3^{8} \cdot 5^{6} \)
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin number field:	Galois closure of 6.2.25920000.1
Galois orbit size:	$1$
Smallest permutation container:	20T145
Parity:	even
Projective image:	$S_6$
Projective field:	Galois closure of 6.2.25920000.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 137 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 137 }$: \( x^{2} + 131x + 3 \)

Roots:

$r_{ 1 }$	$=$	\( 3 + 31\cdot 137 + 3\cdot 137^{2} + 16\cdot 137^{3} + 61\cdot 137^{4} +O(137^{5})\)
$r_{ 2 }$	$=$	\( 115 a + 71 + \left(136 a + 125\right)\cdot 137 + \left(121 a + 96\right)\cdot 137^{2} + \left(69 a + 39\right)\cdot 137^{3} + \left(26 a + 103\right)\cdot 137^{4} +O(137^{5})\)
$r_{ 3 }$	$=$	\( 11 a + 81 + \left(121 a + 18\right)\cdot 137 + \left(26 a + 58\right)\cdot 137^{2} + \left(78 a + 63\right)\cdot 137^{3} + \left(69 a + 121\right)\cdot 137^{4} +O(137^{5})\)
$r_{ 4 }$	$=$	\( 22 a + 76 + 9\cdot 137 + \left(15 a + 7\right)\cdot 137^{2} + \left(67 a + 63\right)\cdot 137^{3} + \left(110 a + 55\right)\cdot 137^{4} +O(137^{5})\)
$r_{ 5 }$	$=$	\( 126 a + 10 + \left(15 a + 49\right)\cdot 137 + \left(110 a + 98\right)\cdot 137^{2} + \left(58 a + 94\right)\cdot 137^{3} + \left(67 a + 49\right)\cdot 137^{4} +O(137^{5})\)
$r_{ 6 }$	$=$	\( 35 + 40\cdot 137 + 10\cdot 137^{2} + 134\cdot 137^{3} + 19\cdot 137^{4} +O(137^{5})\)

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

Size	Order	Action on $r_1, \ldots, r_{ 6 }$	Character values
			$c1$
$1$	$1$	$()$	$9$
$15$	$2$	$(1,2)(3,4)(5,6)$	$-3$
$15$	$2$	$(1,2)$	$-3$
$45$	$2$	$(1,2)(3,4)$	$1$
$40$	$3$	$(1,2,3)(4,5,6)$	$0$
$40$	$3$	$(1,2,3)$	$0$
$90$	$4$	$(1,2,3,4)(5,6)$	$1$
$90$	$4$	$(1,2,3,4)$	$1$
$144$	$5$	$(1,2,3,4,5)$	$-1$
$120$	$6$	$(1,2,3,4,5,6)$	$0$
$120$	$6$	$(1,2,3)(4,5)$	$0$

The blue line marks the conjugacy class containing complex conjugation.