Galois orbit of Artin representations 4.2e6_3e2_13e3.8t21.4

• Label: 4.2e6_3e2_13e3.8t21.4
• Dimension: 4
• Group: $C_2^3 : C_4 $
• Conductor: $ 2^{6} \cdot 3^{2} \cdot 13^{3}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : C_4 $
Conductor:	$1265472= 2^{6} \cdot 3^{2} \cdot 13^{3} $
Artin number field:	Splitting field of $f= x^{8} - 4 x^{7} + 8 x^{6} - 7 x^{4} - 12 x^{3} + 50 x^{2} + 80 x + 28 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$C_2^3: C_4$
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 199 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 24 + 110\cdot 199 + 97\cdot 199^{2} + 76\cdot 199^{3} + 151\cdot 199^{4} + 141\cdot 199^{5} +O\left(199^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 34 + 179\cdot 199 + 9\cdot 199^{2} + 160\cdot 199^{3} + 29\cdot 199^{4} + 14\cdot 199^{5} +O\left(199^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 45 + 124\cdot 199 + 90\cdot 199^{2} + 26\cdot 199^{3} + 165\cdot 199^{4} + 162\cdot 199^{5} +O\left(199^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 58 + 70\cdot 199 + 19\cdot 199^{2} + 34\cdot 199^{3} + 157\cdot 199^{4} + 174\cdot 199^{5} +O\left(199^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 98 + 183\cdot 199 + 135\cdot 199^{3} + 51\cdot 199^{4} + 79\cdot 199^{5} +O\left(199^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 103 + 168\cdot 199 + 52\cdot 199^{2} + 148\cdot 199^{3} + 81\cdot 199^{4} + 14\cdot 199^{5} +O\left(199^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 116 + 35\cdot 199 + 178\cdot 199^{2} + 23\cdot 199^{3} + 18\cdot 199^{4} + 182\cdot 199^{5} +O\left(199^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 123 + 123\cdot 199 + 147\cdot 199^{2} + 191\cdot 199^{3} + 140\cdot 199^{4} + 26\cdot 199^{5} +O\left(199^{ 6 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

Cycle notation

$(2,5)(4,6,8,7)$

$(4,8)(6,7)$

$(1,6,3,7)(2,8,5,4)$

$(1,2)(3,5)(4,7)(6,8)$

$(1,3)(2,5)(4,8)(6,7)$

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 8 }$	Character values
			$c1$
$1$	$1$	$()$	$4$
$1$	$2$	$(1,3)(2,5)(4,8)(6,7)$	$-4$
$2$	$2$	$(1,2)(3,5)(4,7)(6,8)$	$0$
$2$	$2$	$(1,2)(3,5)(4,6)(7,8)$	$0$
$2$	$2$	$(4,8)(6,7)$	$0$
$4$	$2$	$(1,8)(2,6)(3,4)(5,7)$	$0$
$4$	$4$	$(1,6,3,7)(2,8,5,4)$	$0$
$4$	$4$	$(1,8,2,7)(3,4,5,6)$	$0$
$4$	$4$	$(1,7,2,8)(3,6,5,4)$	$0$
$4$	$4$	$(2,5)(4,6,8,7)$	$0$
$4$	$4$	$(2,5)(4,7,8,6)$	$0$

The blue line marks the conjugacy class containing complex conjugation.