Artin representation 4.2e10_3e2_5e2.8t22.6c1

• Label: 4.2e10_3e2_5e2.8t22.6c1
• Dimension: 4
• Group: $C_2^3 : D_4 $
• Conductor: $ 2^{10} \cdot 3^{2} \cdot 5^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : D_4 $
Conductor:	$230400= 2^{10} \cdot 3^{2} \cdot 5^{2} $
Artin number field:	Splitting field of $f= x^{8} + x^{4} + 4 $ 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 5.

Roots:

$r_{ 1 }$	$=$	$ 48 + 196\cdot 241 + 127\cdot 241^{2} + 157\cdot 241^{3} + 160\cdot 241^{4} +O\left(241^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 51 + 99\cdot 241 + 194\cdot 241^{2} + 144\cdot 241^{3} + 115\cdot 241^{4} +O\left(241^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 61 + 132\cdot 241 + 178\cdot 241^{2} + 137\cdot 241^{3} + 194\cdot 241^{4} +O\left(241^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 110 + 128\cdot 241 + 154\cdot 241^{2} + 187\cdot 241^{3} + 109\cdot 241^{4} +O\left(241^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 131 + 112\cdot 241 + 86\cdot 241^{2} + 53\cdot 241^{3} + 131\cdot 241^{4} +O\left(241^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 180 + 108\cdot 241 + 62\cdot 241^{2} + 103\cdot 241^{3} + 46\cdot 241^{4} +O\left(241^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 190 + 141\cdot 241 + 46\cdot 241^{2} + 96\cdot 241^{3} + 125\cdot 241^{4} +O\left(241^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 193 + 44\cdot 241 + 113\cdot 241^{2} + 83\cdot 241^{3} + 80\cdot 241^{4} +O\left(241^{ 5 }\right)$

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

Cycle notation

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

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

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

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.