Artin representation 2.2e6_3e2_5e2.8t7.2c2

• Label: 2.2e6_3e2_5e2.8t7.2c2
• Dimension: 2
• Group: $C_8:C_2$
• Conductor: $ 2^{6} \cdot 3^{2} \cdot 5^{2}$
• Root number: not computed
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$2$
Group:	$C_8:C_2$
Conductor:	$14400= 2^{6} \cdot 3^{2} \cdot 5^{2} $
Artin number field:	Splitting field of $f= x^{8} - 135 x^{4} - 270 x^{2} + 405 $ over $\Q$
Size of Galois orbit:	2
Smallest containing permutation representation:	$C_8:C_2$
Parity:	Odd
Determinant:	1.2e3_5.4t1.2c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 139 }$ to precision 8.

Roots:

$r_{ 1 }$	$=$	$ 29 + 120\cdot 139 + 138\cdot 139^{2} + 55\cdot 139^{3} + 7\cdot 139^{4} + 42\cdot 139^{5} + 84\cdot 139^{6} + 3\cdot 139^{7} +O\left(139^{ 8 }\right)$
$r_{ 2 }$	$=$	$ 38 + 106\cdot 139 + 17\cdot 139^{2} + 44\cdot 139^{3} + 55\cdot 139^{4} + 42\cdot 139^{5} + 38\cdot 139^{6} + 58\cdot 139^{7} +O\left(139^{ 8 }\right)$
$r_{ 3 }$	$=$	$ 41 + 7\cdot 139 + 101\cdot 139^{2} + 137\cdot 139^{3} + 136\cdot 139^{4} + 80\cdot 139^{5} + 33\cdot 139^{6} + 62\cdot 139^{7} +O\left(139^{ 8 }\right)$
$r_{ 4 }$	$=$	$ 64 + 121\cdot 139 + 15\cdot 139^{2} + 18\cdot 139^{3} + 60\cdot 139^{4} + 13\cdot 139^{5} + 40\cdot 139^{6} + 28\cdot 139^{7} +O\left(139^{ 8 }\right)$
$r_{ 5 }$	$=$	$ 75 + 17\cdot 139 + 123\cdot 139^{2} + 120\cdot 139^{3} + 78\cdot 139^{4} + 125\cdot 139^{5} + 98\cdot 139^{6} + 110\cdot 139^{7} +O\left(139^{ 8 }\right)$
$r_{ 6 }$	$=$	$ 98 + 131\cdot 139 + 37\cdot 139^{2} + 139^{3} + 2\cdot 139^{4} + 58\cdot 139^{5} + 105\cdot 139^{6} + 76\cdot 139^{7} +O\left(139^{ 8 }\right)$
$r_{ 7 }$	$=$	$ 101 + 32\cdot 139 + 121\cdot 139^{2} + 94\cdot 139^{3} + 83\cdot 139^{4} + 96\cdot 139^{5} + 100\cdot 139^{6} + 80\cdot 139^{7} +O\left(139^{ 8 }\right)$
$r_{ 8 }$	$=$	$ 110 + 18\cdot 139 + 83\cdot 139^{3} + 131\cdot 139^{4} + 96\cdot 139^{5} + 54\cdot 139^{6} + 135\cdot 139^{7} +O\left(139^{ 8 }\right)$

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

Cycle notation

$(1,8)(4,5)$

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.