Galois orbit of Artin representations 14.283e9_14173e9.30t565.1

• Label: 14.283e9_14173e9.30t565.1
• Dimension: 14
• Group: $S_7$
• Conductor: $ 283^{9} \cdot 14173^{9}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$14$
Group:	$S_7$
Conductor:	$268679173603929851571632658238541005078891351643194468585039= 283^{9} \cdot 14173^{9} $
Artin number field:	Splitting field of $f= x^{7} - 2 x^{6} - 2 x^{5} + 5 x^{4} - 3 x^{3} - 3 x^{2} + 4 x - 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	30T565
Parity:	Odd

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 193 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 193 }$: $ x^{2} + 192 x + 5 $

Roots:

$r_{ 1 }$	$=$	$ 127 a + \left(88 a + 53\right)\cdot 193 + \left(192 a + 177\right)\cdot 193^{2} + \left(77 a + 118\right)\cdot 193^{3} + \left(45 a + 18\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 66 a + 127 + \left(104 a + 14\right)\cdot 193 + 88\cdot 193^{2} + \left(115 a + 4\right)\cdot 193^{3} + \left(147 a + 179\right)\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 27 + 156\cdot 193 + 2\cdot 193^{2} + 175\cdot 193^{3} + 163\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 138 + 63\cdot 193 + 58\cdot 193^{2} + 86\cdot 193^{3} + 170\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 54 + 85\cdot 193 + 51\cdot 193^{2} + 120\cdot 193^{3} + 34\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 19 + 131\cdot 193 + 34\cdot 193^{2} + 134\cdot 193^{3} + 57\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 23 + 75\cdot 193 + 166\cdot 193^{2} + 132\cdot 193^{3} + 147\cdot 193^{4} +O\left(193^{ 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
			$c1$
$1$	$1$	$()$	$14$
$21$	$2$	$(1,2)$	$-4$
$105$	$2$	$(1,2)(3,4)(5,6)$	$0$
$105$	$2$	$(1,2)(3,4)$	$2$
$70$	$3$	$(1,2,3)$	$-1$
$280$	$3$	$(1,2,3)(4,5,6)$	$2$
$210$	$4$	$(1,2,3,4)$	$2$
$630$	$4$	$(1,2,3,4)(5,6)$	$0$
$504$	$5$	$(1,2,3,4,5)$	$-1$
$210$	$6$	$(1,2,3)(4,5)(6,7)$	$-1$
$420$	$6$	$(1,2,3)(4,5)$	$-1$
$840$	$6$	$(1,2,3,4,5,6)$	$0$
$720$	$7$	$(1,2,3,4,5,6,7)$	$0$
$504$	$10$	$(1,2,3,4,5)(6,7)$	$1$
$420$	$12$	$(1,2,3,4)(5,6,7)$	$-1$

The blue line marks the conjugacy class containing complex conjugation.