Galois orbit of Artin representations 4.5e3_101e2.8t21.1

• Label: 4.5e3_101e2.8t21.1
• Dimension: 4
• Group: $C_2^3 : C_4 $
• Conductor: $ 5^{3} \cdot 101^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : C_4 $
Conductor:	$1275125= 5^{3} \cdot 101^{2} $
Artin number field:	Splitting field of $f= x^{8} - 4 x^{7} + 8 x^{6} - 10 x^{5} + 31 x^{4} - 50 x^{3} + 47 x^{2} - 23 x + 6 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$C_2^3: C_4$
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 701 }$ to precision 7.

Roots:

$r_{ 1 }$	$=$	$ 31 + 29\cdot 701 + 368\cdot 701^{2} + 176\cdot 701^{3} + 376\cdot 701^{4} + 215\cdot 701^{5} + 164\cdot 701^{6} +O\left(701^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 254 + 118\cdot 701 + 550\cdot 701^{2} + 255\cdot 701^{3} + 493\cdot 701^{4} + 666\cdot 701^{5} + 652\cdot 701^{6} +O\left(701^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 259 + 321\cdot 701 + 359\cdot 701^{2} + 326\cdot 701^{3} + 440\cdot 701^{4} + 579\cdot 701^{5} + 505\cdot 701^{6} +O\left(701^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 302 + 680\cdot 701 + 190\cdot 701^{2} + 226\cdot 701^{3} + 211\cdot 701^{4} + 321\cdot 701^{5} + 621\cdot 701^{6} +O\left(701^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 400 + 20\cdot 701 + 510\cdot 701^{2} + 474\cdot 701^{3} + 489\cdot 701^{4} + 379\cdot 701^{5} + 79\cdot 701^{6} +O\left(701^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 443 + 379\cdot 701 + 341\cdot 701^{2} + 374\cdot 701^{3} + 260\cdot 701^{4} + 121\cdot 701^{5} + 195\cdot 701^{6} +O\left(701^{ 7 }\right)$
$r_{ 7 }$	$=$	$ 448 + 582\cdot 701 + 150\cdot 701^{2} + 445\cdot 701^{3} + 207\cdot 701^{4} + 34\cdot 701^{5} + 48\cdot 701^{6} +O\left(701^{ 7 }\right)$
$r_{ 8 }$	$=$	$ 671 + 671\cdot 701 + 332\cdot 701^{2} + 524\cdot 701^{3} + 324\cdot 701^{4} + 485\cdot 701^{5} + 536\cdot 701^{6} +O\left(701^{ 7 }\right)$

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

Cycle notation

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

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.