Artin representation 10.7e6_13e6_601e6.30t176.1c1

• Label: 10.7e6_13e6_601e6.30t176.1c1
• Dimension: 10
• Group: $S_6$
• Conductor: $ 7^{6} \cdot 13^{6} \cdot 601^{6}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$10$
Group:	$S_6$
Conductor:	$26760559295546820339317999641= 7^{6} \cdot 13^{6} \cdot 601^{6} $
Artin number field:	Splitting field of $f= x^{6} - x^{5} - 2 x^{3} + 2 x^{2} + 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 $\Q_{ 67 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: $ x^{2} + 63 x + 2 $

Roots:

$r_{ 1 }$	$=$	$ 29 a + 35 + \left(33 a + 45\right)\cdot 67 + \left(a + 45\right)\cdot 67^{2} + \left(38 a + 41\right)\cdot 67^{3} + \left(60 a + 24\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 45 a + 20 + \left(12 a + 3\right)\cdot 67 + \left(9 a + 36\right)\cdot 67^{2} + \left(8 a + 28\right)\cdot 67^{3} + \left(39 a + 20\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 38 a + 17 + \left(33 a + 16\right)\cdot 67 + \left(65 a + 18\right)\cdot 67^{2} + \left(28 a + 58\right)\cdot 67^{3} + \left(6 a + 27\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 10 a + 12 + \left(53 a + 29\right)\cdot 67 + \left(18 a + 9\right)\cdot 67^{2} + \left(23 a + 40\right)\cdot 67^{3} + 57\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 57 a + 52 + \left(13 a + 30\right)\cdot 67 + \left(48 a + 31\right)\cdot 67^{2} + \left(43 a + 47\right)\cdot 67^{3} + \left(66 a + 35\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 22 a + 66 + \left(54 a + 8\right)\cdot 67 + \left(57 a + 60\right)\cdot 67^{2} + \left(58 a + 51\right)\cdot 67^{3} + \left(27 a + 34\right)\cdot 67^{4} +O\left(67^{ 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.