Galois orbit of Artin representations 2.2e4_41e2.8t17.2

• Label: 2.2e4_41e2.8t17.2
• Dimension: 2
• Group: $C_4\wr C_2$
• Conductor: $ 2^{4} \cdot 41^{2}$
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$2$
Group:	$C_4\wr C_2$
Conductor:	$26896= 2^{4} \cdot 41^{2} $
Artin number field:	Splitting field of $f= x^{8} - 4 x^{7} + 8 x^{6} - 10 x^{5} + 11 x^{4} - 6 x^{3} - 4 x^{2} + 4 x + 2 $ over $\Q$
Size of Galois orbit:	2
Smallest containing permutation representation:	$C_4\wr C_2$
Parity:	Odd

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 769 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 66 + 338\cdot 769 + 567\cdot 769^{2} + 583\cdot 769^{3} + 75\cdot 769^{4} + 737\cdot 769^{5} +O\left(769^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 104 + 456\cdot 769 + 442\cdot 769^{2} + 653\cdot 769^{3} + 81\cdot 769^{4} + 340\cdot 769^{5} +O\left(769^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 185 + 276\cdot 769 + 595\cdot 769^{2} + 48\cdot 769^{3} + 155\cdot 769^{4} + 13\cdot 769^{5} +O\left(769^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 297 + 567\cdot 769 + 177\cdot 769^{2} + 161\cdot 769^{3} + 582\cdot 769^{4} + 575\cdot 769^{5} +O\left(769^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 523 + 337\cdot 769 + 553\cdot 769^{2} + 700\cdot 769^{3} + 648\cdot 769^{4} + 348\cdot 769^{5} +O\left(769^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 535 + 356\cdot 769 + 211\cdot 769^{2} + 627\cdot 769^{3} + 151\cdot 769^{4} + 600\cdot 769^{5} +O\left(769^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 642 + 275\cdot 769 + 581\cdot 769^{2} + 165\cdot 769^{3} + 728\cdot 769^{4} + 393\cdot 769^{5} +O\left(769^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 728 + 467\cdot 769 + 715\cdot 769^{2} + 134\cdot 769^{3} + 652\cdot 769^{4} + 66\cdot 769^{5} +O\left(769^{ 6 }\right)$

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

Cycle notation

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

$(2,6,8,4)$

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

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

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

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 8 }$	Character values
			$c1$	$c2$
$1$	$1$	$()$	$2$	$2$
$1$	$2$	$(1,7)(2,8)(3,5)(4,6)$	$-2$	$-2$
$2$	$2$	$(2,8)(4,6)$	$0$	$0$
$4$	$2$	$(1,2)(3,6)(4,5)(7,8)$	$0$	$0$
$1$	$4$	$(1,5,7,3)(2,4,8,6)$	$2 \zeta_{4}$	$-2 \zeta_{4}$
$1$	$4$	$(1,3,7,5)(2,6,8,4)$	$-2 \zeta_{4}$	$2 \zeta_{4}$
$2$	$4$	$(2,6,8,4)$	$\zeta_{4} - 1$	$-\zeta_{4} - 1$
$2$	$4$	$(2,4,8,6)$	$-\zeta_{4} - 1$	$\zeta_{4} - 1$
$2$	$4$	$(1,7)(2,4,8,6)(3,5)$	$-\zeta_{4} + 1$	$\zeta_{4} + 1$
$2$	$4$	$(1,7)(2,6,8,4)(3,5)$	$\zeta_{4} + 1$	$-\zeta_{4} + 1$
$2$	$4$	$(1,5,7,3)(2,6,8,4)$	$0$	$0$
$4$	$4$	$(1,8,7,2)(3,4,5,6)$	$0$	$0$
$4$	$8$	$(1,6,3,8,7,4,5,2)$	$0$	$0$
$4$	$8$	$(1,8,5,6,7,2,3,4)$	$0$	$0$

The blue line marks the conjugacy class containing complex conjugation.