Artin representation 2.2e6_7.8t17.1c2

• Label: 2.2e6_7.8t17.1c2
• Dimension: 2
• Group: $C_4\wr C_2$
• Conductor: $ 2^{6} \cdot 7 $
• Root number: not computed
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$2$
Group:	$C_4\wr C_2$
Conductor:	$448= 2^{6} \cdot 7 $
Artin number field:	Splitting field of $f= x^{8} + 3 x^{6} + 3 x^{4} + 2 x^{2} + 2 $ over $\Q$
Size of Galois orbit:	2
Smallest containing permutation representation:	$C_4\wr C_2$
Parity:	Odd
Determinant:	1.2e4_7.4t1.2c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 127 }$ to precision 7.

Roots:

$r_{ 1 }$	$=$	$ 13 + 18\cdot 127 + 121\cdot 127^{2} + 22\cdot 127^{3} + 60\cdot 127^{4} + 70\cdot 127^{5} + 26\cdot 127^{6} +O\left(127^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 32 + 116\cdot 127 + 49\cdot 127^{2} + 123\cdot 127^{3} + 65\cdot 127^{4} + 93\cdot 127^{5} + 91\cdot 127^{6} +O\left(127^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 34 + 110\cdot 127 + 28\cdot 127^{2} + 59\cdot 127^{3} + 111\cdot 127^{4} + 47\cdot 127^{5} + 121\cdot 127^{6} +O\left(127^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 51 + 17\cdot 127 + 8\cdot 127^{2} + 24\cdot 127^{3} + 123\cdot 127^{4} + 64\cdot 127^{5} + 83\cdot 127^{6} +O\left(127^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 76 + 109\cdot 127 + 118\cdot 127^{2} + 102\cdot 127^{3} + 3\cdot 127^{4} + 62\cdot 127^{5} + 43\cdot 127^{6} +O\left(127^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 93 + 16\cdot 127 + 98\cdot 127^{2} + 67\cdot 127^{3} + 15\cdot 127^{4} + 79\cdot 127^{5} + 5\cdot 127^{6} +O\left(127^{ 7 }\right)$
$r_{ 7 }$	$=$	$ 95 + 10\cdot 127 + 77\cdot 127^{2} + 3\cdot 127^{3} + 61\cdot 127^{4} + 33\cdot 127^{5} + 35\cdot 127^{6} +O\left(127^{ 7 }\right)$
$r_{ 8 }$	$=$	$ 114 + 108\cdot 127 + 5\cdot 127^{2} + 104\cdot 127^{3} + 66\cdot 127^{4} + 56\cdot 127^{5} + 100\cdot 127^{6} +O\left(127^{ 7 }\right)$

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

Cycle notation

$(1,4,8,5)$

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.