Galois orbit of Artin representations 4.2e12_5e3.8t21.2

• Label: 4.2e12_5e3.8t21.2
• Dimension: 4
• Group: $C_2^3 : C_4 $
• Conductor: $ 2^{12} \cdot 5^{3}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : C_4 $
Conductor:	$512000= 2^{12} \cdot 5^{3} $
Artin number field:	Splitting field of $f= x^{8} - 4 x^{7} + 12 x^{6} - 20 x^{5} + 22 x^{4} - 16 x^{3} + 8 x^{2} + 2 $ 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 $\Q_{ 421 }$ to precision 7.

Roots:

$r_{ 1 }$	$=$	$ 94 + 240\cdot 421 + 139\cdot 421^{2} + 316\cdot 421^{3} + 236\cdot 421^{4} + 135\cdot 421^{5} + 16\cdot 421^{6} +O\left(421^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 105 + 13\cdot 421 + 229\cdot 421^{2} + 118\cdot 421^{3} + 226\cdot 421^{4} + 75\cdot 421^{5} + 131\cdot 421^{6} +O\left(421^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 161 + 152\cdot 421 + 108\cdot 421^{2} + 363\cdot 421^{3} + 6\cdot 421^{4} + 328\cdot 421^{5} + 85\cdot 421^{6} +O\left(421^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 183 + 85\cdot 421 + 378\cdot 421^{2} + 319\cdot 421^{3} + 46\cdot 421^{4} + 344\cdot 421^{5} + 153\cdot 421^{6} +O\left(421^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 210 + 306\cdot 421 + 209\cdot 421^{2} + 28\cdot 421^{3} + 405\cdot 421^{4} + 163\cdot 421^{5} + 137\cdot 421^{6} +O\left(421^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 232 + 239\cdot 421 + 58\cdot 421^{2} + 406\cdot 421^{3} + 23\cdot 421^{4} + 180\cdot 421^{5} + 205\cdot 421^{6} +O\left(421^{ 7 }\right)$
$r_{ 7 }$	$=$	$ 346 + 15\cdot 421 + 25\cdot 421^{2} + 375\cdot 421^{3} + 163\cdot 421^{4} + 258\cdot 421^{5} + 419\cdot 421^{6} +O\left(421^{ 7 }\right)$
$r_{ 8 }$	$=$	$ 357 + 209\cdot 421 + 114\cdot 421^{2} + 177\cdot 421^{3} + 153\cdot 421^{4} + 198\cdot 421^{5} + 113\cdot 421^{6} +O\left(421^{ 7 }\right)$

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

Cycle notation

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

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.