Galois orbit of Artin representations 2.2e5_17e2.8t17.1

• Label: 2.2e5_17e2.8t17.1
• Dimension: 2
• Group: $C_4\wr C_2$
• Conductor: $ 2^{5} \cdot 17^{2}$
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$2$
Group:	$C_4\wr C_2$
Conductor:	$9248= 2^{5} \cdot 17^{2} $
Artin number field:	Splitting field of $f= x^{8} - 4 x^{7} + 6 x^{6} - 6 x^{4} + 4 x^{3} + 2 x^{2} + 9 $ 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 $\Q_{ 577 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 62 + 295\cdot 577 + 138\cdot 577^{2} + 28\cdot 577^{3} + 561\cdot 577^{4} +O\left(577^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 284 + 218\cdot 577 + 538\cdot 577^{2} + 155\cdot 577^{3} + 10\cdot 577^{4} +O\left(577^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 291 + 463\cdot 577 + 94\cdot 577^{2} + 385\cdot 577^{3} + 313\cdot 577^{4} +O\left(577^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 375 + 138\cdot 577 + 285\cdot 577^{2} + 568\cdot 577^{3} + 39\cdot 577^{4} +O\left(577^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 433 + 520\cdot 577 + 575\cdot 577^{2} + 548\cdot 577^{3} + 502\cdot 577^{4} +O\left(577^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 438 + 559\cdot 577 + 329\cdot 577^{2} + 261\cdot 577^{3} + 347\cdot 577^{4} +O\left(577^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 481 + 487\cdot 577 + 359\cdot 577^{2} + 350\cdot 577^{3} + 176\cdot 577^{4} +O\left(577^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 525 + 200\cdot 577 + 562\cdot 577^{2} + 8\cdot 577^{3} + 356\cdot 577^{4} +O\left(577^{ 5 }\right)$

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

Cycle notation

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

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

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

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

$(2,7,6,3)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.