Artin representation 2.2e6_5e2.8t17.1c2

• Label: 2.2e6_5e2.8t17.1c2
• Dimension: 2
• Group: $C_4\wr C_2$
• Conductor: $ 2^{6} \cdot 5^{2}$
• Root number: not computed
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$2$
Group:	$C_4\wr C_2$
Conductor:	$1600= 2^{6} \cdot 5^{2} $
Artin number field:	Splitting field of $f= x^{8} + 2 x^{4} + 5 $ over $\Q$
Size of Galois orbit:	2
Smallest containing permutation representation:	$C_4\wr C_2$
Parity:	Odd
Determinant:	1.5.4t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 101 }$ to precision 7.

Roots:

$r_{ 1 }$	$=$	$ 5 + 80\cdot 101 + 6\cdot 101^{2} + 81\cdot 101^{3} + 82\cdot 101^{4} + 64\cdot 101^{5} + 26\cdot 101^{6} +O\left(101^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 27 + 93\cdot 101 + 60\cdot 101^{2} + 4\cdot 101^{3} + 88\cdot 101^{4} + 2\cdot 101^{5} + 33\cdot 101^{6} +O\left(101^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 33 + 43\cdot 101 + 60\cdot 101^{2} + 10\cdot 101^{3} + 95\cdot 101^{4} + 97\cdot 101^{5} + 80\cdot 101^{6} +O\left(101^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 50 + 17\cdot 101 + 7\cdot 101^{2} + 62\cdot 101^{3} + 97\cdot 101^{4} + 64\cdot 101^{5} + 22\cdot 101^{6} +O\left(101^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 51 + 83\cdot 101 + 93\cdot 101^{2} + 38\cdot 101^{3} + 3\cdot 101^{4} + 36\cdot 101^{5} + 78\cdot 101^{6} +O\left(101^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 68 + 57\cdot 101 + 40\cdot 101^{2} + 90\cdot 101^{3} + 5\cdot 101^{4} + 3\cdot 101^{5} + 20\cdot 101^{6} +O\left(101^{ 7 }\right)$
$r_{ 7 }$	$=$	$ 74 + 7\cdot 101 + 40\cdot 101^{2} + 96\cdot 101^{3} + 12\cdot 101^{4} + 98\cdot 101^{5} + 67\cdot 101^{6} +O\left(101^{ 7 }\right)$
$r_{ 8 }$	$=$	$ 96 + 20\cdot 101 + 94\cdot 101^{2} + 19\cdot 101^{3} + 18\cdot 101^{4} + 36\cdot 101^{5} + 74\cdot 101^{6} +O\left(101^{ 7 }\right)$

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

Cycle notation

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

$(2,3,7,6)$

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.