Artin representation 2.928.8t17.a.a

• Label: 2.928.8t17.a.a
• Dimension: $2$
• Group: $C_4\wr C_2$
• Conductor: $928$
• Root number: not computed
• Indicator: $0$

Basic invariants

Dimension:	$2$
Group:	$C_4\wr C_2$
Conductor:	$928$$\medspace = 2^{5} \cdot 29$
Artin stem field:	Galois closure of 8.0.1598357504.1
Galois orbit size:	$2$
Smallest permutation container:	$C_4\wr C_2$
Parity:	odd
Determinant:	1.29.4t1.a.b
Projective image:	$D_4$
Projective stem field:	Galois closure of 4.2.390224.1

Defining polynomial

$f(x)$

$=$

$x^{8} - 10x^{4} + 29$ .

The roots of $f$ are computed in $\Q_{ 181 }$ to precision 8.

Roots:

$r_{ 1 }$	$=$	$32 + 150\cdot 181 + 33\cdot 181^{2} + 98\cdot 181^{3} + 4\cdot 181^{5} + 120\cdot 181^{6} + 54\cdot 181^{7} +O(181^{8})$
$r_{ 2 }$	$=$	$53 + 110\cdot 181 + 40\cdot 181^{2} + 142\cdot 181^{3} + 62\cdot 181^{4} + 130\cdot 181^{5} + 117\cdot 181^{6} + 58\cdot 181^{7} +O(181^{8})$
$r_{ 3 }$	$=$	$65 + 22\cdot 181 + 64\cdot 181^{2} + 163\cdot 181^{3} + 53\cdot 181^{4} + 88\cdot 181^{5} + 167\cdot 181^{6} + 82\cdot 181^{7} +O(181^{8})$
$r_{ 4 }$	$=$	$79 + 155\cdot 181 + 56\cdot 181^{2} + 96\cdot 181^{3} + 167\cdot 181^{4} + 37\cdot 181^{5} + 120\cdot 181^{6} + 26\cdot 181^{7} +O(181^{8})$
$r_{ 5 }$	$=$	$102 + 25\cdot 181 + 124\cdot 181^{2} + 84\cdot 181^{3} + 13\cdot 181^{4} + 143\cdot 181^{5} + 60\cdot 181^{6} + 154\cdot 181^{7} +O(181^{8})$
$r_{ 6 }$	$=$	$116 + 158\cdot 181 + 116\cdot 181^{2} + 17\cdot 181^{3} + 127\cdot 181^{4} + 92\cdot 181^{5} + 13\cdot 181^{6} + 98\cdot 181^{7} +O(181^{8})$
$r_{ 7 }$	$=$	$128 + 70\cdot 181 + 140\cdot 181^{2} + 38\cdot 181^{3} + 118\cdot 181^{4} + 50\cdot 181^{5} + 63\cdot 181^{6} + 122\cdot 181^{7} +O(181^{8})$
$r_{ 8 }$	$=$	$149 + 30\cdot 181 + 147\cdot 181^{2} + 82\cdot 181^{3} + 180\cdot 181^{4} + 176\cdot 181^{5} + 60\cdot 181^{6} + 126\cdot 181^{7} +O(181^{8})$

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

Cycle notation

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

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

$(2,4,7,5)$

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

$(2,7)(4,5)$

Character values on conjugacy classes

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