Artin representation 2.2e2_97.8t17.1c2

• Label: 2.2e2_97.8t17.1c2
• Dimension: 2
• Group: $C_4\wr C_2$
• Conductor: $ 2^{2} \cdot 97 $
• Root number: not computed
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$2$
Group:	$C_4\wr C_2$
Conductor:	$388= 2^{2} \cdot 97 $
Artin number field:	Splitting field of $f= x^{8} - 4 x^{7} + 8 x^{6} - 10 x^{5} + 13 x^{4} - 10 x^{3} - 4 x^{2} + 6 x + 5 $ over $\Q$
Size of Galois orbit:	2
Smallest containing permutation representation:	$C_4\wr C_2$
Parity:	Odd
Determinant:	1.2e2_97.4t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 241 }$ to precision 7.

Roots:

$r_{ 1 }$	$=$	$ 68 + 74\cdot 241 + 20\cdot 241^{2} + 46\cdot 241^{3} + 121\cdot 241^{4} + 161\cdot 241^{5} + 78\cdot 241^{6} +O\left(241^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 73 + 195\cdot 241 + 13\cdot 241^{2} + 69\cdot 241^{3} + 31\cdot 241^{4} + 70\cdot 241^{5} + 75\cdot 241^{6} +O\left(241^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 94 + 155\cdot 241 + 111\cdot 241^{2} + 180\cdot 241^{3} + 58\cdot 241^{4} + 168\cdot 241^{5} + 12\cdot 241^{6} +O\left(241^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 110 + 104\cdot 241 + 30\cdot 241^{2} + 234\cdot 241^{3} + 120\cdot 241^{4} + 222\cdot 241^{5} + 143\cdot 241^{6} +O\left(241^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 191 + 231\cdot 241 + 77\cdot 241^{2} + 217\cdot 241^{3} + 106\cdot 241^{4} + 232\cdot 241^{5} + 110\cdot 241^{6} +O\left(241^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 212 + 147\cdot 241 + 78\cdot 241^{2} + 21\cdot 241^{3} + 181\cdot 241^{4} + 170\cdot 241^{5} + 5\cdot 241^{6} +O\left(241^{ 7 }\right)$
$r_{ 7 }$	$=$	$ 228 + 187\cdot 241 + 213\cdot 241^{2} + 62\cdot 241^{3} + 135\cdot 241^{4} + 151\cdot 241^{5} + 111\cdot 241^{6} +O\left(241^{ 7 }\right)$
$r_{ 8 }$	$=$	$ 233 + 107\cdot 241 + 176\cdot 241^{2} + 132\cdot 241^{3} + 208\cdot 241^{4} + 27\cdot 241^{5} + 184\cdot 241^{6} +O\left(241^{ 7 }\right)$

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

Cycle notation

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

$(2,8)(3,6)$

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

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

$(2,3,8,6)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.