Artin representation 10.5e4_103e4_127e4.30t176.1c1

• Label: 10.5e4_103e4_127e4.30t176.1c1
• Dimension: 10
• Group: $S_6$
• Conductor: $ 5^{4} \cdot 103^{4} \cdot 127^{4}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$10$
Group:	$S_6$
Conductor:	$18299692832486700625= 5^{4} \cdot 103^{4} \cdot 127^{4} $
Artin number field:	Splitting field of $f= x^{6} - x^{5} + 2 x^{4} - 2 x^{3} - 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	30T176
Parity:	Even
Determinant:	1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $ x^{2} + 29 x + 3 $

Roots:

$r_{ 1 }$	$=$	$ 20 a + 12 + \left(19 a + 4\right)\cdot 31 + \left(28 a + 14\right)\cdot 31^{2} + \left(30 a + 30\right)\cdot 31^{3} + \left(8 a + 7\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 11 a + 21 + \left(11 a + 23\right)\cdot 31 + \left(2 a + 20\right)\cdot 31^{2} + 31^{3} + \left(22 a + 26\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 8 a + 30 + \left(14 a + 16\right)\cdot 31 + \left(13 a + 12\right)\cdot 31^{2} + 27\cdot 31^{3} + \left(12 a + 23\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 19 a + 20 + \left(21 a + 8\right)\cdot 31 + \left(15 a + 5\right)\cdot 31^{2} + 31^{3} + \left(8 a + 17\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 23 a + 15 + \left(16 a + 6\right)\cdot 31 + \left(17 a + 25\right)\cdot 31^{2} + \left(30 a + 14\right)\cdot 31^{3} + \left(18 a + 16\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 12 a + 27 + \left(9 a + 1\right)\cdot 31 + \left(15 a + 15\right)\cdot 31^{2} + \left(30 a + 17\right)\cdot 31^{3} + \left(22 a + 1\right)\cdot 31^{4} +O\left(31^{ 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

Size	Order	Action on $r_1, \ldots, r_{ 6 }$	Character value
$1$	$1$	$()$	$10$
$15$	$2$	$(1,2)(3,4)(5,6)$	$-2$
$15$	$2$	$(1,2)$	$2$
$45$	$2$	$(1,2)(3,4)$	$-2$
$40$	$3$	$(1,2,3)(4,5,6)$	$1$
$40$	$3$	$(1,2,3)$	$1$
$90$	$4$	$(1,2,3,4)(5,6)$	$0$
$90$	$4$	$(1,2,3,4)$	$0$
$144$	$5$	$(1,2,3,4,5)$	$0$
$120$	$6$	$(1,2,3,4,5,6)$	$1$
$120$	$6$	$(1,2,3)(4,5)$	$-1$

The blue line marks the conjugacy class containing complex conjugation.