Galois orbit of Artin representations 4.2e12_5e2.8t15.1

• Label: 4.2e12_5e2.8t15.1
• Dimension: 4
• Group: $Z_8 : Z_8^\times$
• Conductor: $ 2^{12} \cdot 5^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$Z_8 : Z_8^\times$
Conductor:	$102400= 2^{12} \cdot 5^{2} $
Artin number field:	Splitting field of $f= x^{8} - 4 x^{6} + 5 x^{4} - 2 x^{2} - 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 $\Q_{ 521 }$ to precision 7.

Roots:

$r_{ 1 }$	$=$	$ 58 + 453\cdot 521 + 284\cdot 521^{2} + 103\cdot 521^{3} + 311\cdot 521^{4} + 314\cdot 521^{5} + 191\cdot 521^{6} +O\left(521^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 90 + 200\cdot 521 + 257\cdot 521^{2} + 57\cdot 521^{3} + 402\cdot 521^{4} + 280\cdot 521^{5} + 324\cdot 521^{6} +O\left(521^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 184 + 227\cdot 521 + 194\cdot 521^{2} + 94\cdot 521^{3} + 228\cdot 521^{4} + 106\cdot 521^{5} + 146\cdot 521^{6} +O\left(521^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 191 + 497\cdot 521 + 360\cdot 521^{2} + 143\cdot 521^{3} + 104\cdot 521^{4} + 125\cdot 521^{5} + 301\cdot 521^{6} +O\left(521^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 330 + 23\cdot 521 + 160\cdot 521^{2} + 377\cdot 521^{3} + 416\cdot 521^{4} + 395\cdot 521^{5} + 219\cdot 521^{6} +O\left(521^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 337 + 293\cdot 521 + 326\cdot 521^{2} + 426\cdot 521^{3} + 292\cdot 521^{4} + 414\cdot 521^{5} + 374\cdot 521^{6} +O\left(521^{ 7 }\right)$
$r_{ 7 }$	$=$	$ 431 + 320\cdot 521 + 263\cdot 521^{2} + 463\cdot 521^{3} + 118\cdot 521^{4} + 240\cdot 521^{5} + 196\cdot 521^{6} +O\left(521^{ 7 }\right)$
$r_{ 8 }$	$=$	$ 463 + 67\cdot 521 + 236\cdot 521^{2} + 417\cdot 521^{3} + 209\cdot 521^{4} + 206\cdot 521^{5} + 329\cdot 521^{6} +O\left(521^{ 7 }\right)$

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

Cycle notation

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

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.