Galois orbit of Artin representations 2.3e2_37e2.8t17.1

• Label: 2.3e2_37e2.8t17.1
• Dimension: 2
• Group: $C_4\wr C_2$
• Conductor: $ 3^{2} \cdot 37^{2}$
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$2$
Group:	$C_4\wr C_2$
Conductor:	$12321= 3^{2} \cdot 37^{2} $
Artin number field:	Splitting field of $f= x^{8} - x^{7} + 6 x^{6} - x^{5} + 10 x^{4} + 6 x^{3} + 6 x^{2} + 9 x + 3 $ 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 $\Q_{ 223 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 20 + 113\cdot 223 + 54\cdot 223^{2} + 184\cdot 223^{3} + 126\cdot 223^{4} +O\left(223^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 95 + 75\cdot 223 + 67\cdot 223^{2} + 137\cdot 223^{3} + 115\cdot 223^{4} +O\left(223^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 118 + 21\cdot 223 + 164\cdot 223^{2} + 80\cdot 223^{3} + 109\cdot 223^{4} +O\left(223^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 132 + 183\cdot 223^{2} + 182\cdot 223^{3} + 119\cdot 223^{4} +O\left(223^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 146 + 188\cdot 223 + 101\cdot 223^{3} + 195\cdot 223^{4} +O\left(223^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 189 + 155\cdot 223 + 67\cdot 223^{2} + 153\cdot 223^{3} + 43\cdot 223^{4} +O\left(223^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 202 + 9\cdot 223 + 31\cdot 223^{2} + 4\cdot 223^{3} + 16\cdot 223^{4} +O\left(223^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 214 + 103\cdot 223 + 100\cdot 223^{2} + 48\cdot 223^{3} + 165\cdot 223^{4} +O\left(223^{ 5 }\right)$

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

Cycle notation

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

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

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

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

$(2,6)(4,8)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.