Galois orbit of Artin representations 4.3e2_13e4.8t16.2

• Label: 4.3e2_13e4.8t16.2
• Dimension: 4
• Group: $(C_8:C_2):C_2$
• Conductor: $ 3^{2} \cdot 13^{4}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$(C_8:C_2):C_2$
Conductor:	$257049= 3^{2} \cdot 13^{4} $
Artin number field:	Splitting field of $f= x^{8} - 4 x^{7} + 7 x^{6} - 7 x^{5} + 6 x^{4} - 5 x^{3} - 2 x^{2} + 4 x + 3 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$(C_8:C_2):C_2$
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 367 }$ to precision 8.

Roots:

$r_{ 1 }$	$=$	$ 53 + 253\cdot 367 + 250\cdot 367^{2} + 330\cdot 367^{3} + 45\cdot 367^{4} + 280\cdot 367^{5} + 71\cdot 367^{6} + 140\cdot 367^{7} +O\left(367^{ 8 }\right)$
$r_{ 2 }$	$=$	$ 95 + 214\cdot 367 + 361\cdot 367^{2} + 254\cdot 367^{3} + 163\cdot 367^{4} + 186\cdot 367^{5} + 56\cdot 367^{6} + 267\cdot 367^{7} +O\left(367^{ 8 }\right)$
$r_{ 3 }$	$=$	$ 118 + 224\cdot 367 + 28\cdot 367^{2} + 191\cdot 367^{3} + 53\cdot 367^{4} + 17\cdot 367^{5} + 179\cdot 367^{6} + 209\cdot 367^{7} +O\left(367^{ 8 }\right)$
$r_{ 4 }$	$=$	$ 167 + 242\cdot 367 + 198\cdot 367^{2} + 220\cdot 367^{3} + 313\cdot 367^{4} + 60\cdot 367^{5} + 218\cdot 367^{6} + 271\cdot 367^{7} +O\left(367^{ 8 }\right)$
$r_{ 5 }$	$=$	$ 201 + 124\cdot 367 + 168\cdot 367^{2} + 146\cdot 367^{3} + 53\cdot 367^{4} + 306\cdot 367^{5} + 148\cdot 367^{6} + 95\cdot 367^{7} +O\left(367^{ 8 }\right)$
$r_{ 6 }$	$=$	$ 250 + 142\cdot 367 + 338\cdot 367^{2} + 175\cdot 367^{3} + 313\cdot 367^{4} + 349\cdot 367^{5} + 187\cdot 367^{6} + 157\cdot 367^{7} +O\left(367^{ 8 }\right)$
$r_{ 7 }$	$=$	$ 273 + 152\cdot 367 + 5\cdot 367^{2} + 112\cdot 367^{3} + 203\cdot 367^{4} + 180\cdot 367^{5} + 310\cdot 367^{6} + 99\cdot 367^{7} +O\left(367^{ 8 }\right)$
$r_{ 8 }$	$=$	$ 315 + 113\cdot 367 + 116\cdot 367^{2} + 36\cdot 367^{3} + 321\cdot 367^{4} + 86\cdot 367^{5} + 295\cdot 367^{6} + 226\cdot 367^{7} +O\left(367^{ 8 }\right)$

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

Cycle notation

$(3,6)(4,5)$

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

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

$(2,7)(3,6)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.