Galois orbit of Artin representations 4.5e4_11e2_41e2.8t16.9

• Label: 4.5e4_11e2_41e2.8t16.9
• Dimension: 4
• Group: $(C_8:C_2):C_2$
• Conductor: $ 5^{4} \cdot 11^{2} \cdot 41^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$(C_8:C_2):C_2$
Conductor:	$127125625= 5^{4} \cdot 11^{2} \cdot 41^{2} $
Artin number field:	Splitting field of $f= x^{8} - 2 x^{7} - 27 x^{6} + 41 x^{5} + 315 x^{4} - 344 x^{3} - 1747 x^{2} + 1073 x + 4331 $ 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 $\Q_{ 661 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 22 + 324\cdot 661 + 280\cdot 661^{2} + 368\cdot 661^{3} + 57\cdot 661^{4} + 37\cdot 661^{5} +O\left(661^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 237 + 365\cdot 661 + 18\cdot 661^{2} + 437\cdot 661^{3} + 104\cdot 661^{4} + 247\cdot 661^{5} +O\left(661^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 274 + 636\cdot 661 + 659\cdot 661^{2} + 21\cdot 661^{3} + 66\cdot 661^{4} + 133\cdot 661^{5} +O\left(661^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 380 + 252\cdot 661 + 623\cdot 661^{2} + 636\cdot 661^{3} + 629\cdot 661^{4} + 584\cdot 661^{5} +O\left(661^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 423 + 313\cdot 661 + 379\cdot 661^{2} + 407\cdot 661^{3} + 548\cdot 661^{4} + 398\cdot 661^{5} +O\left(661^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 431 + 467\cdot 661 + 477\cdot 661^{2} + 398\cdot 661^{3} + 281\cdot 661^{4} + 582\cdot 661^{5} +O\left(661^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 432 + 67\cdot 661 + 20\cdot 661^{2} + 226\cdot 661^{3} + 521\cdot 661^{4} + 356\cdot 661^{5} +O\left(661^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 447 + 216\cdot 661 + 184\cdot 661^{2} + 147\cdot 661^{3} + 434\cdot 661^{4} + 303\cdot 661^{5} +O\left(661^{ 6 }\right)$

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

Cycle notation

$(2,7)(6,8)$

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

$(3,4)(6,8)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.