Galois orbit of Artin representations 4.2e12_13e2.8t15.2

• Label: 4.2e12_13e2.8t15.2
• Dimension: 4
• Group: $Z_8 : Z_8^\times$
• Conductor: $ 2^{12} \cdot 13^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$Z_8 : Z_8^\times$
Conductor:	$692224= 2^{12} \cdot 13^{2} $
Artin number field:	Splitting field of $f= x^{8} - 3 x^{4} - 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 $\Q_{ 433 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 16 + 75\cdot 433 + 34\cdot 433^{2} + 204\cdot 433^{3} + 110\cdot 433^{4} + 123\cdot 433^{5} +O\left(433^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 32 + 296\cdot 433 + 236\cdot 433^{2} + 79\cdot 433^{3} + 104\cdot 433^{4} + 229\cdot 433^{5} +O\left(433^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 99 + 370\cdot 433 + 208\cdot 433^{2} + 253\cdot 433^{3} + 238\cdot 433^{4} + 422\cdot 433^{5} +O\left(433^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 167 + 108\cdot 433 + 274\cdot 433^{2} + 158\cdot 433^{3} + 277\cdot 433^{4} + 204\cdot 433^{5} +O\left(433^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 266 + 324\cdot 433 + 158\cdot 433^{2} + 274\cdot 433^{3} + 155\cdot 433^{4} + 228\cdot 433^{5} +O\left(433^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 334 + 62\cdot 433 + 224\cdot 433^{2} + 179\cdot 433^{3} + 194\cdot 433^{4} + 10\cdot 433^{5} +O\left(433^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 401 + 136\cdot 433 + 196\cdot 433^{2} + 353\cdot 433^{3} + 328\cdot 433^{4} + 203\cdot 433^{5} +O\left(433^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 417 + 357\cdot 433 + 398\cdot 433^{2} + 228\cdot 433^{3} + 322\cdot 433^{4} + 309\cdot 433^{5} +O\left(433^{ 6 }\right)$

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

Cycle notation

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

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.