Galois orbit of Artin representations 2.2e2_17e2.8t17.2

• Label: 2.2e2_17e2.8t17.2
• Dimension: 2
• Group: $C_4\wr C_2$
• Conductor: $ 2^{2} \cdot 17^{2}$
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$2$
Group:	$C_4\wr C_2$
Conductor:	$1156= 2^{2} \cdot 17^{2} $
Artin number field:	Splitting field of $f= x^{8} - 2 x^{7} + 10 x^{5} - 6 x^{4} - 20 x^{3} + 55 x^{2} - 62 x + 25 $ 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 $\Q_{ 149 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 21 + 4\cdot 149 + 2\cdot 149^{2} + 88\cdot 149^{3} + 123\cdot 149^{4} +O\left(149^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 32 + 74\cdot 149 + 107\cdot 149^{2} + 109\cdot 149^{3} + 61\cdot 149^{4} +O\left(149^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 35 + 80\cdot 149 + 87\cdot 149^{2} + 24\cdot 149^{3} + 40\cdot 149^{4} +O\left(149^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 41 + 25\cdot 149 + 29\cdot 149^{2} + 132\cdot 149^{3} + 117\cdot 149^{4} +O\left(149^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 86 + 106\cdot 149 + 107\cdot 149^{2} + 123\cdot 149^{3} + 64\cdot 149^{4} +O\left(149^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 92 + 113\cdot 149 + 41\cdot 149^{2} + 103\cdot 149^{3} + 5\cdot 149^{4} +O\left(149^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 143 + 66\cdot 149 + 44\cdot 149^{2} + 91\cdot 149^{3} + 12\cdot 149^{4} +O\left(149^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 148 + 124\cdot 149 + 26\cdot 149^{2} + 72\cdot 149^{3} + 20\cdot 149^{4} +O\left(149^{ 5 }\right)$

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

Cycle notation

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

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

$(2,5,4,3)$

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.