Galois orbit of Artin representations 4.2e11_3e2_7e2.8t21.2

• Label: 4.2e11_3e2_7e2.8t21.2
• Dimension: 4
• Group: $C_2^3 : C_4 $
• Conductor: $ 2^{11} \cdot 3^{2} \cdot 7^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : C_4 $
Conductor:	$903168= 2^{11} \cdot 3^{2} \cdot 7^{2} $
Artin number field:	Splitting field of $f= x^{8} - 2 x^{6} + 2 x^{4} + 20 x^{2} + 16 $ 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 $\Q_{ 617 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 168 + 226\cdot 617 + 111\cdot 617^{2} + 255\cdot 617^{3} + 295\cdot 617^{4} + 487\cdot 617^{5} +O\left(617^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 172 + 339\cdot 617 + 199\cdot 617^{2} + 402\cdot 617^{3} + 439\cdot 617^{4} + 436\cdot 617^{5} +O\left(617^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 175 + 239\cdot 617 + 158\cdot 617^{2} + 538\cdot 617^{3} + 334\cdot 617^{4} + 10\cdot 617^{5} +O\left(617^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 187 + 231\cdot 617 + 117\cdot 617^{2} + 349\cdot 617^{3} + 65\cdot 617^{4} + 511\cdot 617^{5} +O\left(617^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 430 + 385\cdot 617 + 499\cdot 617^{2} + 267\cdot 617^{3} + 551\cdot 617^{4} + 105\cdot 617^{5} +O\left(617^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 442 + 377\cdot 617 + 458\cdot 617^{2} + 78\cdot 617^{3} + 282\cdot 617^{4} + 606\cdot 617^{5} +O\left(617^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 445 + 277\cdot 617 + 417\cdot 617^{2} + 214\cdot 617^{3} + 177\cdot 617^{4} + 180\cdot 617^{5} +O\left(617^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 449 + 390\cdot 617 + 505\cdot 617^{2} + 361\cdot 617^{3} + 321\cdot 617^{4} + 129\cdot 617^{5} +O\left(617^{ 6 }\right)$

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

Cycle notation

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

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.