Artin representation 4.2e10_3e2_7e2.8t22.4c1

• Label: 4.2e10_3e2_7e2.8t22.4c1
• Dimension: 4
• Group: $C_2^3 : D_4 $
• Conductor: $ 2^{10} \cdot 3^{2} \cdot 7^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : D_4 $
Conductor:	$451584= 2^{10} \cdot 3^{2} \cdot 7^{2} $
Artin number field:	Splitting field of $f= x^{8} - 7 x^{6} + 17 x^{4} - 14 x^{2} + 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 $\Q_{ 193 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 9 + 36\cdot 193 + 74\cdot 193^{2} + 176\cdot 193^{3} + 117\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 25 + 150\cdot 193 + 17\cdot 193^{2} + 129\cdot 193^{3} + 65\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 80 + 103\cdot 193^{2} + 177\cdot 193^{3} + 58\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 87 + 57\cdot 193 + 186\cdot 193^{2} + 87\cdot 193^{3} + 79\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 106 + 135\cdot 193 + 6\cdot 193^{2} + 105\cdot 193^{3} + 113\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 113 + 192\cdot 193 + 89\cdot 193^{2} + 15\cdot 193^{3} + 134\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 168 + 42\cdot 193 + 175\cdot 193^{2} + 63\cdot 193^{3} + 127\cdot 193^{4} +O\left(193^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 184 + 156\cdot 193 + 118\cdot 193^{2} + 16\cdot 193^{3} + 75\cdot 193^{4} +O\left(193^{ 5 }\right)$

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

Cycle notation

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

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

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

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.