Galois orbit of Artin representations 4.2e12_7e2.8t15.2

• Label: 4.2e12_7e2.8t15.2
• Dimension: 4
• Group: $Z_8 : Z_8^\times$
• Conductor: $ 2^{12} \cdot 7^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$Z_8 : Z_8^\times$
Conductor:	$200704= 2^{12} \cdot 7^{2} $
Artin number field:	Splitting field of $f= x^{8} - 4 x^{5} + 8 x^{3} + 4 x^{2} + 4 x + 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$Z_8 : Z_8^\times$
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 281 }$ to precision 8.

Roots:

$r_{ 1 }$	$=$	$ 25 + 76\cdot 281 + 216\cdot 281^{2} + 165\cdot 281^{3} + 95\cdot 281^{4} + 195\cdot 281^{5} + 37\cdot 281^{6} + 93\cdot 281^{7} +O\left(281^{ 8 }\right)$
$r_{ 2 }$	$=$	$ 44 + 272\cdot 281 + 251\cdot 281^{2} + 149\cdot 281^{3} + 22\cdot 281^{4} + 152\cdot 281^{5} + 56\cdot 281^{6} + 72\cdot 281^{7} +O\left(281^{ 8 }\right)$
$r_{ 3 }$	$=$	$ 88 + 211\cdot 281 + 242\cdot 281^{2} + 23\cdot 281^{3} + 207\cdot 281^{4} + 9\cdot 281^{5} + 106\cdot 281^{6} + 109\cdot 281^{7} +O\left(281^{ 8 }\right)$
$r_{ 4 }$	$=$	$ 124 + 2\cdot 281 + 132\cdot 281^{2} + 222\cdot 281^{3} + 236\cdot 281^{4} + 204\cdot 281^{5} + 80\cdot 281^{6} + 6\cdot 281^{7} +O\left(281^{ 8 }\right)$
$r_{ 5 }$	$=$	$ 173 + 66\cdot 281 + 66\cdot 281^{2} + 184\cdot 281^{3} + 62\cdot 281^{4} + 2\cdot 281^{5} + 153\cdot 281^{6} + 63\cdot 281^{7} +O\left(281^{ 8 }\right)$
$r_{ 6 }$	$=$	$ 211 + 90\cdot 281 + 94\cdot 281^{2} + 32\cdot 281^{3} + 95\cdot 281^{4} + 240\cdot 281^{5} + 172\cdot 281^{6} + 231\cdot 281^{7} +O\left(281^{ 8 }\right)$
$r_{ 7 }$	$=$	$ 219 + 268\cdot 281 + 253\cdot 281^{2} + 74\cdot 281^{3} + 237\cdot 281^{4} + 159\cdot 281^{5} + 226\cdot 281^{6} + 148\cdot 281^{7} +O\left(281^{ 8 }\right)$
$r_{ 8 }$	$=$	$ 240 + 135\cdot 281 + 147\cdot 281^{2} + 270\cdot 281^{3} + 166\cdot 281^{4} + 159\cdot 281^{5} + 9\cdot 281^{6} + 118\cdot 281^{7} +O\left(281^{ 8 }\right)$

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

Cycle notation

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

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

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

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

$(1,4)(6,7)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.