Artin representation 4.3e3_13e2_97e2.8t29.1c1

• Label: 4.3e3_13e2_97e2.8t29.1c1
• Dimension: 4
• Group: $(((C_4 \times C_2): C_2):C_2):C_2$
• Conductor: $ 3^{3} \cdot 13^{2} \cdot 97^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$(((C_4 \times C_2): C_2):C_2):C_2$
Conductor:	$42933267= 3^{3} \cdot 13^{2} \cdot 97^{2} $
Artin number field:	Splitting field of $f= x^{8} - x^{7} - 8 x^{6} + x^{5} + 14 x^{4} + 22 x^{3} + 20 x^{2} - 35 x - 22 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$(((C_4 \times C_2): C_2):C_2):C_2$
Parity:	Odd
Determinant:	1.3.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 103 }$ to precision 10.

Roots:

$r_{ 1 }$	$=$	$ 10 + 61\cdot 103 + 68\cdot 103^{2} + 85\cdot 103^{3} + 80\cdot 103^{4} + 29\cdot 103^{5} + 30\cdot 103^{6} + 23\cdot 103^{7} + 32\cdot 103^{8} + 11\cdot 103^{9} +O\left(103^{ 10 }\right)$
$r_{ 2 }$	$=$	$ 20 + 28\cdot 103 + 98\cdot 103^{2} + 62\cdot 103^{3} + 96\cdot 103^{4} + 52\cdot 103^{5} + 69\cdot 103^{6} + 47\cdot 103^{7} + 103^{8} + 66\cdot 103^{9} +O\left(103^{ 10 }\right)$
$r_{ 3 }$	$=$	$ 25 + 75\cdot 103 + 73\cdot 103^{2} + 75\cdot 103^{3} + 31\cdot 103^{4} + 18\cdot 103^{5} + 35\cdot 103^{6} + 12\cdot 103^{7} + 84\cdot 103^{8} + 26\cdot 103^{9} +O\left(103^{ 10 }\right)$
$r_{ 4 }$	$=$	$ 53 + 47\cdot 103 + 5\cdot 103^{2} + 10\cdot 103^{3} + 68\cdot 103^{4} + 102\cdot 103^{5} + 48\cdot 103^{6} + 53\cdot 103^{7} + 52\cdot 103^{8} + 45\cdot 103^{9} +O\left(103^{ 10 }\right)$
$r_{ 5 }$	$=$	$ 64 + 44\cdot 103 + 75\cdot 103^{2} + 17\cdot 103^{3} + 46\cdot 103^{4} + 60\cdot 103^{5} + 3\cdot 103^{6} + 44\cdot 103^{7} + 46\cdot 103^{8} + 101\cdot 103^{9} +O\left(103^{ 10 }\right)$
$r_{ 6 }$	$=$	$ 77 + 11\cdot 103 + 9\cdot 103^{2} + 45\cdot 103^{3} + 41\cdot 103^{4} + 31\cdot 103^{5} + 98\cdot 103^{6} + 40\cdot 103^{7} + 55\cdot 103^{8} + 87\cdot 103^{9} +O\left(103^{ 10 }\right)$
$r_{ 7 }$	$=$	$ 80 + 62\cdot 103 + 28\cdot 103^{2} + 56\cdot 103^{3} + 87\cdot 103^{4} + 54\cdot 103^{5} + 26\cdot 103^{6} + 25\cdot 103^{7} + 4\cdot 103^{8} + 103^{9} +O\left(103^{ 10 }\right)$
$r_{ 8 }$	$=$	$ 84 + 80\cdot 103 + 52\cdot 103^{2} + 58\cdot 103^{3} + 62\cdot 103^{4} + 61\cdot 103^{5} + 99\cdot 103^{6} + 61\cdot 103^{7} + 32\cdot 103^{8} + 72\cdot 103^{9} +O\left(103^{ 10 }\right)$

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

Cycle notation

$(1,2)(3,8)$

$(2,3)(4,7)$

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

$(4,7)(5,6)$

$(1,8)(4,7)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.