Galois orbit of Artin representations 20.3e10_31e10_470149e10.70.1

• Label: 20.3e10_31e10_470149e10.70.1
• Dimension: 20
• Group: $S_7$
• Conductor: $ 3^{10} \cdot 31^{10} \cdot 470149^{10}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$20$
Group:	$S_7$
Conductor:	$25537868867782213785576187807596629714815503008391029318218826158767781257249= 3^{10} \cdot 31^{10} \cdot 470149^{10} $
Artin number field:	Splitting field of $f= x^{7} - x^{6} - 7 x^{5} + 7 x^{4} + 12 x^{3} - 10 x^{2} - 6 x + 3 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	70
Parity:	Even

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} + 131 x + 3 $

Roots:

$r_{ 1 }$	$=$	$ 107 + 128\cdot 137 + 60\cdot 137^{2} + 58\cdot 137^{3} +O\left(137^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 126 a + 18 + \left(29 a + 91\right)\cdot 137 + \left(71 a + 89\right)\cdot 137^{2} + \left(88 a + 46\right)\cdot 137^{3} + \left(42 a + 28\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 19 + 31\cdot 137 + 13\cdot 137^{2} + 41\cdot 137^{3} + 57\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 64 + 85\cdot 137 + 119\cdot 137^{2} + 63\cdot 137^{3} + 14\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 29 + 69\cdot 137 + 99\cdot 137^{2} + 81\cdot 137^{3} + 93\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 86 + 134\cdot 137 + 88\cdot 137^{2} + 23\cdot 137^{3} + 21\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 11 a + 89 + \left(107 a + 7\right)\cdot 137 + \left(65 a + 76\right)\cdot 137^{2} + \left(48 a + 95\right)\cdot 137^{3} + \left(94 a + 58\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 7 }$

Cycle notation

$(1,2,3,4,5,6,7)$

$(1,2)$

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 7 }$	Character values
			$c1$
$1$	$1$	$()$	$20$
$21$	$2$	$(1,2)$	$0$
$105$	$2$	$(1,2)(3,4)(5,6)$	$0$
$105$	$2$	$(1,2)(3,4)$	$-4$
$70$	$3$	$(1,2,3)$	$2$
$280$	$3$	$(1,2,3)(4,5,6)$	$2$
$210$	$4$	$(1,2,3,4)$	$0$
$630$	$4$	$(1,2,3,4)(5,6)$	$0$
$504$	$5$	$(1,2,3,4,5)$	$0$
$210$	$6$	$(1,2,3)(4,5)(6,7)$	$2$
$420$	$6$	$(1,2,3)(4,5)$	$0$
$840$	$6$	$(1,2,3,4,5,6)$	$0$
$720$	$7$	$(1,2,3,4,5,6,7)$	$-1$
$504$	$10$	$(1,2,3,4,5)(6,7)$	$0$
$420$	$12$	$(1,2,3,4)(5,6,7)$	$0$

The blue line marks the conjugacy class containing complex conjugation.