Galois orbit of Artin representations 4.3e2_5e2_7e2_11e2.8t22.3

• Label: 4.3e2_5e2_7e2_11e2.8t22.3
• Dimension: 4
• Group: $C_2^3 : D_4 $
• Conductor: $ 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 11^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : D_4 $
Conductor:	$1334025= 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 11^{2} $
Artin number field:	Splitting field of $f= x^{8} - 3 x^{7} + 6 x^{6} - x^{5} + 19 x^{4} + 9 x^{3} + 22 x^{2} + 7 x + 7 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$C_2^3 : D_4 $
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 331 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 23 + 103\cdot 331 + 162\cdot 331^{2} + 278\cdot 331^{3} + 331^{4} +O\left(331^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 39 + 239\cdot 331 + 242\cdot 331^{2} + 236\cdot 331^{3} + 83\cdot 331^{4} +O\left(331^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 107 + 114\cdot 331 + 263\cdot 331^{2} + 136\cdot 331^{3} + 163\cdot 331^{4} +O\left(331^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 136 + 208\cdot 331 + 146\cdot 331^{2} + 189\cdot 331^{3} + 315\cdot 331^{4} +O\left(331^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 223 + 93\cdot 331 + 151\cdot 331^{2} + 329\cdot 331^{3} + 240\cdot 331^{4} +O\left(331^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 234 + 57\cdot 331 + 263\cdot 331^{2} + 268\cdot 331^{3} + 131\cdot 331^{4} +O\left(331^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 267 + 112\cdot 331 + 203\cdot 331^{2} + 271\cdot 331^{3} + 243\cdot 331^{4} +O\left(331^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 298 + 63\cdot 331 + 222\cdot 331^{2} + 274\cdot 331^{3} + 142\cdot 331^{4} +O\left(331^{ 5 }\right)$

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

Cycle notation

$(1,3)(2,6)$

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

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

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

$(1,3)(7,8)$

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

$(1,3)(4,5)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.