Galois orbit of Artin representations 2.2e2_17e2.8t17.1

• Label: 2.2e2_17e2.8t17.1
• 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} + 4 x^{5} - 4 x^{4} + 3 x^{2} - 2 x + 1 $ 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:

$r_{ 1 }$	$=$	$ 23 + 19\cdot 149 + 46\cdot 149^{2} + 81\cdot 149^{3} + 103\cdot 149^{4} +O\left(149^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 61 + 34\cdot 149 + 129\cdot 149^{2} + 79\cdot 149^{3} + 110\cdot 149^{4} +O\left(149^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 81 + 8\cdot 149 + 91\cdot 149^{2} + 83\cdot 149^{3} + 71\cdot 149^{4} +O\left(149^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 98 + 18\cdot 149 + 67\cdot 149^{3} + 149^{4} +O\left(149^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 100 + 63\cdot 149 + 100\cdot 149^{2} + 70\cdot 149^{3} + 78\cdot 149^{4} +O\left(149^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 122 + 35\cdot 149 + 36\cdot 149^{2} + 22\cdot 149^{3} + 101\cdot 149^{4} +O\left(149^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 124 + 114\cdot 149 + 49\cdot 149^{2} + 142\cdot 149^{3} + 36\cdot 149^{4} +O\left(149^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 138 + 2\cdot 149 + 143\cdot 149^{2} + 48\cdot 149^{3} + 92\cdot 149^{4} +O\left(149^{ 5 }\right)$

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

Cycle notation

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

$(2,3)(7,8)$

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

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

$(2,7,3,8)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.