Galois orbit of Artin representations 10.11850950346603023041.30t164.a

• Label: 10.118...041.30t164.a
• Dimension: $10$
• Group: $S_6$
• Conductor: $1.185\times 10^{19}$
• Indicator: $1$

Basic invariants

Dimension:	$10$
Group:	$S_6$
Conductor:	\(11850950346603023041\)\(\medspace = 23^{4} \cdot 2551^{4} \)
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin number field:	Galois closure of 6.2.58673.1
Galois orbit size:	$1$
Smallest permutation container:	30T164
Parity:	even
Projective image:	$S_6$
Projective field:	Galois closure of 6.2.58673.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: \( x^{2} + 49x + 2 \)

Roots:

$r_{ 1 }$	$=$	\( 19 a + 10 + \left(19 a + 45\right)\cdot 53 + \left(26 a + 21\right)\cdot 53^{2} + \left(12 a + 29\right)\cdot 53^{3} + \left(44 a + 28\right)\cdot 53^{4} +O(53^{5})\)
$r_{ 2 }$	$=$	\( 28 + 50\cdot 53 + 33\cdot 53^{2} + 15\cdot 53^{3} + 25\cdot 53^{4} +O(53^{5})\)
$r_{ 3 }$	$=$	\( 6 + 48\cdot 53 + 28\cdot 53^{2} + 12\cdot 53^{3} + 22\cdot 53^{4} +O(53^{5})\)
$r_{ 4 }$	$=$	\( 24 + 25\cdot 53 + 15\cdot 53^{2} + 14\cdot 53^{3} + 36\cdot 53^{4} +O(53^{5})\)
$r_{ 5 }$	$=$	\( 34 a + 33 + \left(33 a + 50\right)\cdot 53 + \left(26 a + 1\right)\cdot 53^{2} + 40 a\cdot 53^{3} + \left(8 a + 34\right)\cdot 53^{4} +O(53^{5})\)
$r_{ 6 }$	$=$	\( 5 + 45\cdot 53 + 3\cdot 53^{2} + 34\cdot 53^{3} + 12\cdot 53^{4} +O(53^{5})\)

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
			$c1$
$1$	$1$	$()$	$10$
$15$	$2$	$(1,2)(3,4)(5,6)$	$-2$
$15$	$2$	$(1,2)$	$2$
$45$	$2$	$(1,2)(3,4)$	$-2$
$40$	$3$	$(1,2,3)(4,5,6)$	$1$
$40$	$3$	$(1,2,3)$	$1$
$90$	$4$	$(1,2,3,4)(5,6)$	$0$
$90$	$4$	$(1,2,3,4)$	$0$
$144$	$5$	$(1,2,3,4,5)$	$0$
$120$	$6$	$(1,2,3,4,5,6)$	$1$
$120$	$6$	$(1,2,3)(4,5)$	$-1$

The blue line marks the conjugacy class containing complex conjugation.