Artin representation 4.2e4_3e2_7e2_13e2.8t22.10c1

• Label: 4.2e4_3e2_7e2_13e2.8t22.10c1
• Dimension: 4
• Group: $C_2^3 : D_4 $
• Conductor: $ 2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 13^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : D_4 $
Conductor:	$1192464= 2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 13^{2} $
Artin number field:	Splitting field of $f= x^{8} - 5 x^{6} - 8 x^{5} - 4 x^{4} + 26 x^{3} + 60 x^{2} + 62 x + 37 $ 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 $\Q_{ 337 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 35 + 154\cdot 337 + 336\cdot 337^{2} + 240\cdot 337^{3} + 134\cdot 337^{4} +O\left(337^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 82 + 114\cdot 337 + 59\cdot 337^{2} + 217\cdot 337^{3} + 154\cdot 337^{4} +O\left(337^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 98 + 11\cdot 337 + 156\cdot 337^{2} + 335\cdot 337^{3} + 117\cdot 337^{4} +O\left(337^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 122 + 57\cdot 337 + 122\cdot 337^{2} + 217\cdot 337^{3} + 266\cdot 337^{4} +O\left(337^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 130 + 272\cdot 337 + 29\cdot 337^{2} + 314\cdot 337^{3} + 18\cdot 337^{4} +O\left(337^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 135 + 88\cdot 337 + 147\cdot 337^{2} + 126\cdot 337^{3} + 267\cdot 337^{4} +O\left(337^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 152 + 6\cdot 337 + 323\cdot 337^{2} + 16\cdot 337^{3} + 217\cdot 337^{4} +O\left(337^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 257 + 306\cdot 337 + 173\cdot 337^{2} + 216\cdot 337^{3} + 170\cdot 337^{4} +O\left(337^{ 5 }\right)$

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

Cycle notation

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

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

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

$(1,2)(5,6)$

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.