Galois orbit of Artin representations 4.2e4_3e2_5e2_7e2.8t22.10

• Label: 4.2e4_3e2_5e2_7e2.8t22.10
• Dimension: 4
• Group: $C_2^3 : D_4 $
• Conductor: $ 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : D_4 $
Conductor:	$176400= 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} $
Artin number field:	Splitting field of $f= x^{8} - 2 x^{7} + 4 x^{6} - 10 x^{5} + 9 x^{4} - 14 x^{3} + 16 x^{2} - 4 x + 13 $ 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 $\Q_{ 421 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 77 + 194\cdot 421 + 213\cdot 421^{2} + 99\cdot 421^{3} + 286\cdot 421^{4} +O\left(421^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 91 + 97\cdot 421 + 148\cdot 421^{2} + 75\cdot 421^{3} + 175\cdot 421^{4} +O\left(421^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 106 + 306\cdot 421 + 38\cdot 421^{2} + 156\cdot 421^{3} + 107\cdot 421^{4} +O\left(421^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 177 + 273\cdot 421 + 274\cdot 421^{2} + 162\cdot 421^{3} + 242\cdot 421^{4} +O\left(421^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 198 + 407\cdot 421 + 195\cdot 421^{2} + 319\cdot 421^{3} + 143\cdot 421^{4} +O\left(421^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 279 + 273\cdot 421 + 383\cdot 421^{2} + 276\cdot 421^{3} + 392\cdot 421^{4} +O\left(421^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 355 + 389\cdot 421 + 349\cdot 421^{2} + 281\cdot 421^{3} + 274\cdot 421^{4} +O\left(421^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 403 + 162\cdot 421 + 79\cdot 421^{2} + 312\cdot 421^{3} + 61\cdot 421^{4} +O\left(421^{ 5 }\right)$

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

Cycle notation

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

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

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

$(5,8)(6,7)$

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.