Galois orbit of Artin representations 4.2e4_3e2_5e2_7e2.8t22.5

• Label: 4.2e4_3e2_5e2_7e2.8t22.5
• 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} - 3 x^{7} - x^{6} + 10 x^{5} - 20 x^{3} + 9 x^{2} + 7 x + 1 $ 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:

$r_{ 1 }$	$=$	$ 35 + 265\cdot 421 + 18\cdot 421^{2} + 29\cdot 421^{3} + 187\cdot 421^{4} +O\left(421^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 47 + 113\cdot 421 + 270\cdot 421^{2} + 159\cdot 421^{3} + 121\cdot 421^{4} +O\left(421^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 62 + 64\cdot 421 + 368\cdot 421^{2} + 400\cdot 421^{3} + 213\cdot 421^{4} +O\left(421^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 81 + 39\cdot 421 + 373\cdot 421^{2} + 307\cdot 421^{3} + 245\cdot 421^{4} +O\left(421^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 134 + 155\cdot 421 + 145\cdot 421^{2} + 397\cdot 421^{3} + 135\cdot 421^{4} +O\left(421^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 248 + 325\cdot 421 + 125\cdot 421^{2} + 273\cdot 421^{3} + 184\cdot 421^{4} +O\left(421^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 280 + 157\cdot 421 + 53\cdot 421^{2} + 189\cdot 421^{3} + 144\cdot 421^{4} +O\left(421^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 379 + 142\cdot 421 + 329\cdot 421^{2} + 347\cdot 421^{3} + 29\cdot 421^{4} +O\left(421^{ 5 }\right)$

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

Cycle notation

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

$(1,4)(3,5)$

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

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.