Artin representation 2.2e2_17e2.8t7.1c2

• Label: 2.2e2_17e2.8t7.1c2
• Dimension: 2
• Group: $C_8:C_2$
• Conductor: $ 2^{2} \cdot 17^{2}$
• Root number: not computed
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$2$
Group:	$C_8:C_2$
Conductor:	$1156= 2^{2} \cdot 17^{2} $
Artin number field:	Splitting field of $f= x^{8} - 3 x^{7} + 5 x^{6} + 9 x^{5} - 26 x^{4} + 35 x^{3} - 31 x^{2} + 10 x - 1 $ over $\Q$
Size of Galois orbit:	2
Smallest containing permutation representation:	$C_8:C_2$
Parity:	Odd
Determinant:	1.2e2_17.4t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 89 }$ to precision 7.

Roots:

$r_{ 1 }$	$=$	$ 4 + 66\cdot 89 + 77\cdot 89^{2} + 37\cdot 89^{3} + 32\cdot 89^{4} + 73\cdot 89^{5} +O\left(89^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 35 + 7\cdot 89 + 88\cdot 89^{2} + 9\cdot 89^{3} + 79\cdot 89^{4} + 23\cdot 89^{5} + 77\cdot 89^{6} +O\left(89^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 52 + 2\cdot 89 + 19\cdot 89^{2} + 33\cdot 89^{3} + 59\cdot 89^{4} + 40\cdot 89^{5} + 87\cdot 89^{6} +O\left(89^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 57 + 33\cdot 89 + 82\cdot 89^{2} + 30\cdot 89^{3} + 33\cdot 89^{4} + 38\cdot 89^{5} + 85\cdot 89^{6} +O\left(89^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 67 + 58\cdot 89 + 58\cdot 89^{2} + 79\cdot 89^{3} + 30\cdot 89^{4} + 35\cdot 89^{5} + 65\cdot 89^{6} +O\left(89^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 73 + 68\cdot 89 + 87\cdot 89^{2} + 80\cdot 89^{3} + 49\cdot 89^{4} + 37\cdot 89^{5} + 82\cdot 89^{6} +O\left(89^{ 7 }\right)$
$r_{ 7 }$	$=$	$ 79 + 59\cdot 89 + 83\cdot 89^{2} + 43\cdot 89^{3} + 47\cdot 89^{4} + 58\cdot 89^{5} + 6\cdot 89^{6} +O\left(89^{ 7 }\right)$
$r_{ 8 }$	$=$	$ 81 + 58\cdot 89 + 36\cdot 89^{2} + 39\cdot 89^{3} + 23\cdot 89^{4} + 48\cdot 89^{5} + 39\cdot 89^{6} +O\left(89^{ 7 }\right)$

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

Cycle notation

$(3,5)(4,7)$

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

$(1,8)(2,6)$

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.