Galois orbit of Artin representations 4.2e12_3e2_7e2.8t22.5

• Label: 4.2e12_3e2_7e2.8t22.5
• Dimension: 4
• Group: $C_2^3 : D_4 $
• Conductor: $ 2^{12} \cdot 3^{2} \cdot 7^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : D_4 $
Conductor:	$1806336= 2^{12} \cdot 3^{2} \cdot 7^{2} $
Artin number field:	Splitting field of $f= x^{8} - 4 x^{6} - 12 x^{5} + 22 x^{4} + 16 x^{3} + 8 x^{2} - 80 x + 50 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$C_2^3 : D_4 $
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 337 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 34 + 93\cdot 337^{2} + 186\cdot 337^{3} + 308\cdot 337^{4} + 267\cdot 337^{5} +O\left(337^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 162 + 194\cdot 337 + 297\cdot 337^{2} + 257\cdot 337^{3} + 3\cdot 337^{4} + 313\cdot 337^{5} +O\left(337^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 176 + 14\cdot 337 + 121\cdot 337^{2} + 125\cdot 337^{3} + 239\cdot 337^{4} + 183\cdot 337^{5} +O\left(337^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 184 + 29\cdot 337 + 256\cdot 337^{2} + 247\cdot 337^{3} + 40\cdot 337^{4} + 320\cdot 337^{5} +O\left(337^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 224 + 322\cdot 337 + 294\cdot 337^{2} + 38\cdot 337^{3} + 64\cdot 337^{4} + 178\cdot 337^{5} +O\left(337^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 294 + 112\cdot 337 + 27\cdot 337^{2} + 319\cdot 337^{3} + 320\cdot 337^{4} + 109\cdot 337^{5} +O\left(337^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 302 + 127\cdot 337 + 162\cdot 337^{2} + 104\cdot 337^{3} + 122\cdot 337^{4} + 246\cdot 337^{5} +O\left(337^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 309 + 208\cdot 337 + 95\cdot 337^{2} + 68\cdot 337^{3} + 248\cdot 337^{4} + 65\cdot 337^{5} +O\left(337^{ 6 }\right)$

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

Cycle notation

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

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

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

$(1,2)(3,7)$

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.