Galois orbit of Artin representations 4.5e3_101.8t27.1

• Label: 4.5e3_101.8t27.1
• Dimension: 4
• Group: $((C_8 : C_2):C_2):C_2$
• Conductor: $ 5^{3} \cdot 101 $
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$((C_8 : C_2):C_2):C_2$
Conductor:	$12625= 5^{3} \cdot 101 $
Artin number field:	Splitting field of $f= x^{8} - 2 x^{7} + 3 x^{5} - x^{4} - 3 x^{3} + 2 x + 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$((C_8 : C_2):C_2):C_2$
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 701 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 144 + 248\cdot 701 + 197\cdot 701^{2} + 198\cdot 701^{3} + 233\cdot 701^{4} + 228\cdot 701^{5} +O\left(701^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 165 + 624\cdot 701 + 174\cdot 701^{2} + 575\cdot 701^{3} + 106\cdot 701^{4} + 612\cdot 701^{5} +O\left(701^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 200 + 53\cdot 701 + 412\cdot 701^{2} + 676\cdot 701^{3} + 358\cdot 701^{4} + 525\cdot 701^{5} +O\left(701^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 258 + 378\cdot 701 + 392\cdot 701^{2} + 481\cdot 701^{3} + 397\cdot 701^{4} + 610\cdot 701^{5} +O\left(701^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 354 + 477\cdot 701 + 167\cdot 701^{2} + 127\cdot 701^{3} + 142\cdot 701^{4} + 672\cdot 701^{5} +O\left(701^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 424 + 501\cdot 701 + 511\cdot 701^{2} + 304\cdot 701^{3} + 430\cdot 701^{4} + 462\cdot 701^{5} +O\left(701^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 577 + 273\cdot 701 + 300\cdot 701^{2} + 417\cdot 701^{3} + 340\cdot 701^{4} + 100\cdot 701^{5} +O\left(701^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 684 + 246\cdot 701 + 647\cdot 701^{2} + 22\cdot 701^{3} + 93\cdot 701^{4} + 293\cdot 701^{5} +O\left(701^{ 6 }\right)$

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

Cycle notation

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

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

$(6,7)$

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

$(2,8)(6,7)$

$(3,5)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.