Galois orbit of Artin representations 4.5e4_151e2.8t16.1

• Label: 4.5e4_151e2.8t16.1
• Dimension: 4
• Group: $(C_8:C_2):C_2$
• Conductor: $ 5^{4} \cdot 151^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$(C_8:C_2):C_2$
Conductor:	$14250625= 5^{4} \cdot 151^{2} $
Artin number field:	Splitting field of $f= x^{8} - 2 x^{7} + 18 x^{6} - 39 x^{5} + 140 x^{4} - 219 x^{3} + 453 x^{2} - 382 x + 451 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$(C_8:C_2):C_2$
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 311 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 17 + 261\cdot 311 + 225\cdot 311^{2} + 110\cdot 311^{3} + 119\cdot 311^{4} + 99\cdot 311^{5} +O\left(311^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 26 + 282\cdot 311 + 302\cdot 311^{2} + 12\cdot 311^{3} + 31\cdot 311^{4} + 137\cdot 311^{5} +O\left(311^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 32 + 201\cdot 311 + 305\cdot 311^{2} + 34\cdot 311^{3} + 234\cdot 311^{4} + 245\cdot 311^{5} +O\left(311^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 35 + 261\cdot 311 + 167\cdot 311^{2} + 106\cdot 311^{3} + 123\cdot 311^{4} + 148\cdot 311^{5} +O\left(311^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 37 + 133\cdot 311 + 91\cdot 311^{2} + 76\cdot 311^{3} + 49\cdot 311^{4} + 247\cdot 311^{5} +O\left(311^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 57 + 269\cdot 311 + 75\cdot 311^{2} + 65\cdot 311^{3} + 229\cdot 311^{4} + 112\cdot 311^{5} +O\left(311^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 188 + 201\cdot 311 + 72\cdot 311^{2} + 104\cdot 311^{3} + 35\cdot 311^{4} + 115\cdot 311^{5} +O\left(311^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 232 + 256\cdot 311 + 311^{2} + 111\cdot 311^{3} + 111\cdot 311^{4} + 138\cdot 311^{5} +O\left(311^{ 6 }\right)$

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

Cycle notation

$(3,6)(4,7)$

$(1,2)(3,6)$

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

$(4,7)(5,8)$

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

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 8 }$	Character values
			$c1$
$1$	$1$	$()$	$4$
$1$	$2$	$(1,2)(3,6)(4,7)(5,8)$	$-4$
$2$	$2$	$(3,6)(4,7)$	$0$
$4$	$2$	$(1,2)(3,6)$	$0$
$4$	$2$	$(1,5)(2,8)(3,4)(6,7)$	$0$
$2$	$4$	$(1,5,2,8)(3,4,6,7)$	$0$
$2$	$4$	$(1,5,2,8)(3,7,6,4)$	$0$
$4$	$8$	$(1,4,8,3,2,7,5,6)$	$0$
$4$	$8$	$(1,3,5,4,2,6,8,7)$	$0$
$4$	$8$	$(1,7,5,6,2,4,8,3)$	$0$
$4$	$8$	$(1,6,8,7,2,3,5,4)$	$0$

The blue line marks the conjugacy class containing complex conjugation.