Artin representation 4.2e10_3e4_5e2.8t22.5c1

• Label: 4.2e10_3e4_5e2.8t22.5c1
• Dimension: 4
• Group: $C_2^3 : D_4 $
• Conductor: $ 2^{10} \cdot 3^{4} \cdot 5^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : D_4 $
Conductor:	$2073600= 2^{10} \cdot 3^{4} \cdot 5^{2} $
Artin number field:	Splitting field of $f= x^{8} - 6 x^{6} + 27 x^{4} - 54 x^{2} + 36 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$C_2^3 : D_4 $
Parity:	Even
Determinant:	1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 241 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 11 + 133\cdot 241 + 12\cdot 241^{2} + 67\cdot 241^{3} + 81\cdot 241^{4} + 134\cdot 241^{5} +O\left(241^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 28 + 99\cdot 241 + 117\cdot 241^{2} + 32\cdot 241^{3} + 80\cdot 241^{4} + 168\cdot 241^{5} +O\left(241^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 72 + 13\cdot 241 + 192\cdot 241^{2} + 186\cdot 241^{3} + 81\cdot 241^{4} + 29\cdot 241^{5} +O\left(241^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 119 + 21\cdot 241 + 150\cdot 241^{2} + 129\cdot 241^{3} + 128\cdot 241^{4} + 173\cdot 241^{5} +O\left(241^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 122 + 219\cdot 241 + 90\cdot 241^{2} + 111\cdot 241^{3} + 112\cdot 241^{4} + 67\cdot 241^{5} +O\left(241^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 169 + 227\cdot 241 + 48\cdot 241^{2} + 54\cdot 241^{3} + 159\cdot 241^{4} + 211\cdot 241^{5} +O\left(241^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 213 + 141\cdot 241 + 123\cdot 241^{2} + 208\cdot 241^{3} + 160\cdot 241^{4} + 72\cdot 241^{5} +O\left(241^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 230 + 107\cdot 241 + 228\cdot 241^{2} + 173\cdot 241^{3} + 159\cdot 241^{4} + 106\cdot 241^{5} +O\left(241^{ 6 }\right)$

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

Cycle notation

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

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

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

$(1,8)(4,5)$

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.