Artin representation 4.2e4_13e2_17e3.8t21.7c1

• Label: 4.2e4_13e2_17e3.8t21.7c1
• Dimension: 4
• Group: $C_2^3 : C_4 $
• Conductor: $ 2^{4} \cdot 13^{2} \cdot 17^{3}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : C_4 $
Conductor:	$13284752= 2^{4} \cdot 13^{2} \cdot 17^{3} $
Artin number field:	Splitting field of $f= x^{8} - 5 x^{6} - 11 x^{4} - 48 x^{3} + 36 x^{2} + 240 x + 212 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$C_2^3: C_4$
Parity:	Even
Determinant:	1.17.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 157 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 37 + 57\cdot 157 + 98\cdot 157^{2} + 141\cdot 157^{3} + 56\cdot 157^{4} + 68\cdot 157^{5} +O\left(157^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 79 + 156\cdot 157 + 98\cdot 157^{2} + 148\cdot 157^{3} + 25\cdot 157^{4} + 7\cdot 157^{5} +O\left(157^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 86 + 26\cdot 157 + 10\cdot 157^{2} + 84\cdot 157^{3} + 114\cdot 157^{4} + 86\cdot 157^{5} +O\left(157^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 87 + 66\cdot 157 + 121\cdot 157^{2} + 73\cdot 157^{3} + 134\cdot 157^{4} + 45\cdot 157^{5} +O\left(157^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 91 + 40\cdot 157 + 131\cdot 157^{2} + 108\cdot 157^{3} + 60\cdot 157^{4} + 51\cdot 157^{5} +O\left(157^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 115 + 112\cdot 157 + 121\cdot 157^{2} + 59\cdot 157^{3} + 139\cdot 157^{4} + 42\cdot 157^{5} +O\left(157^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 139 + 103\cdot 157 + 22\cdot 157^{2} + 64\cdot 157^{3} + 95\cdot 157^{4} + 102\cdot 157^{5} +O\left(157^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 151 + 63\cdot 157 + 23\cdot 157^{2} + 104\cdot 157^{3} + 66\cdot 157^{5} +O\left(157^{ 6 }\right)$

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

Cycle notation

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

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

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

$(3,6)(5,8)$

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.