Galois orbit of Artin representations 4.2e8_7e4.8t15.1

• Label: 4.2e8_7e4.8t15.1
• Dimension: 4
• Group: $Z_8 : Z_8^\times$
• Conductor: $ 2^{8} \cdot 7^{4}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$Z_8 : Z_8^\times$
Conductor:	$614656= 2^{8} \cdot 7^{4} $
Artin number field:	Splitting field of $f= x^{8} - 2 x^{7} + 7 x^{4} - 14 x^{3} + 12 x + 4 $ 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 $\Q_{ 337 }$ to precision 8.

Roots:

$r_{ 1 }$	$=$	$ 17 + 213\cdot 337 + 123\cdot 337^{2} + 263\cdot 337^{3} + 33\cdot 337^{4} + 189\cdot 337^{5} + 289\cdot 337^{6} + 280\cdot 337^{7} +O\left(337^{ 8 }\right)$
$r_{ 2 }$	$=$	$ 60 + 316\cdot 337 + 2\cdot 337^{2} + 193\cdot 337^{3} + 42\cdot 337^{4} + 159\cdot 337^{5} + 99\cdot 337^{6} + 21\cdot 337^{7} +O\left(337^{ 8 }\right)$
$r_{ 3 }$	$=$	$ 108 + 316\cdot 337 + 151\cdot 337^{2} + 324\cdot 337^{3} + 191\cdot 337^{4} + 184\cdot 337^{5} + 249\cdot 337^{6} + 255\cdot 337^{7} +O\left(337^{ 8 }\right)$
$r_{ 4 }$	$=$	$ 153 + 165\cdot 337 + 58\cdot 337^{2} + 230\cdot 337^{3} + 68\cdot 337^{4} + 141\cdot 337^{5} + 35\cdot 337^{6} + 116\cdot 337^{7} +O\left(337^{ 8 }\right)$
$r_{ 5 }$	$=$	$ 219 + 109\cdot 337 + 157\cdot 337^{2} + 44\cdot 337^{3} + 25\cdot 337^{4} + 92\cdot 337^{5} + 129\cdot 337^{6} + 283\cdot 337^{7} +O\left(337^{ 8 }\right)$
$r_{ 6 }$	$=$	$ 223 + 260\cdot 337 + 23\cdot 337^{2} + 187\cdot 337^{3} + 209\cdot 337^{4} + 156\cdot 337^{5} + 254\cdot 337^{6} + 266\cdot 337^{7} +O\left(337^{ 8 }\right)$
$r_{ 7 }$	$=$	$ 243 + 82\cdot 337 + 118\cdot 337^{2} + 206\cdot 337^{3} + 200\cdot 337^{4} + 281\cdot 337^{5} + 72\cdot 337^{6} + 253\cdot 337^{7} +O\left(337^{ 8 }\right)$
$r_{ 8 }$	$=$	$ 327 + 220\cdot 337 + 37\cdot 337^{2} + 236\cdot 337^{3} + 238\cdot 337^{4} + 143\cdot 337^{5} + 217\cdot 337^{6} + 207\cdot 337^{7} +O\left(337^{ 8 }\right)$

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

Cycle notation

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

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

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

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

$(2,4)(6,8)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.