Galois orbit of Artin representations 2.3_7_13e2.8t17.2

• Label: 2.3_7_13e2.8t17.2
• Dimension: 2
• Group: $C_4\wr C_2$
• Conductor: $ 3 \cdot 7 \cdot 13^{2}$
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$2$
Group:	$C_4\wr C_2$
Conductor:	$3549= 3 \cdot 7 \cdot 13^{2} $
Artin number field:	Splitting field of $f= x^{8} - 3 x^{7} + 5 x^{5} + x^{4} - 3 x^{3} - 5 x^{2} - 2 x + 7 $ over $\Q$
Size of Galois orbit:	2
Smallest containing permutation representation:	$C_4\wr C_2$
Parity:	Odd

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 103 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 7 + 97\cdot 103 + 86\cdot 103^{2} + 66\cdot 103^{3} + 43\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 10 + 51\cdot 103 + 32\cdot 103^{2} + 35\cdot 103^{3} + 79\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 18 + 34\cdot 103 + 62\cdot 103^{2} + 66\cdot 103^{3} + 55\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 19 + 51\cdot 103 + 83\cdot 103^{2} + 89\cdot 103^{3} + 43\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 41 + 98\cdot 103 + 68\cdot 103^{2} + 28\cdot 103^{3} + 100\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 44 + 2\cdot 103 + 14\cdot 103^{2} + 22\cdot 103^{3} + 6\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 81 + 38\cdot 103 + 102\cdot 103^{2} + 80\cdot 103^{3} + 17\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 92 + 38\cdot 103 + 64\cdot 103^{2} + 21\cdot 103^{3} + 65\cdot 103^{4} +O\left(103^{ 5 }\right)$

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

Cycle notation

$(2,5,7,4)$

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

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

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

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

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 8 }$	Character values
			$c1$	$c2$
$1$	$1$	$()$	$2$	$2$
$1$	$2$	$(1,3)(2,7)(4,5)(6,8)$	$-2$	$-2$
$2$	$2$	$(2,7)(4,5)$	$0$	$0$
$4$	$2$	$(1,2)(3,7)(4,6)(5,8)$	$0$	$0$
$1$	$4$	$(1,6,3,8)(2,4,7,5)$	$2 \zeta_{4}$	$-2 \zeta_{4}$
$1$	$4$	$(1,8,3,6)(2,5,7,4)$	$-2 \zeta_{4}$	$2 \zeta_{4}$
$2$	$4$	$(2,5,7,4)$	$\zeta_{4} - 1$	$-\zeta_{4} - 1$
$2$	$4$	$(2,4,7,5)$	$-\zeta_{4} - 1$	$\zeta_{4} - 1$
$2$	$4$	$(1,3)(2,4,7,5)(6,8)$	$-\zeta_{4} + 1$	$\zeta_{4} + 1$
$2$	$4$	$(1,3)(2,5,7,4)(6,8)$	$\zeta_{4} + 1$	$-\zeta_{4} + 1$
$2$	$4$	$(1,6,3,8)(2,5,7,4)$	$0$	$0$
$4$	$4$	$(1,7,3,2)(4,6,5,8)$	$0$	$0$
$4$	$8$	$(1,5,8,7,3,4,6,2)$	$0$	$0$
$4$	$8$	$(1,7,6,5,3,2,8,4)$	$0$	$0$

The blue line marks the conjugacy class containing complex conjugation.