Galois orbit of Artin representations 4.3e4_37e3.8t21.1

• Label: 4.3e4_37e3.8t21.1
• Dimension: 4
• Group: $C_2^3 : C_4 $
• Conductor: $ 3^{4} \cdot 37^{3}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : C_4 $
Conductor:	$4102893= 3^{4} \cdot 37^{3} $
Artin number field:	Splitting field of $f= x^{8} - 3 x^{7} + 5 x^{6} - 15 x^{5} + 18 x^{4} - 9 x^{3} + 47 x^{2} + 54 x + 13 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$C_2^3: C_4$
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 229 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 70 + 164\cdot 229 + 189\cdot 229^{2} + 132\cdot 229^{3} + 122\cdot 229^{4} + 68\cdot 229^{5} +O\left(229^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 153 + 72\cdot 229 + 149\cdot 229^{2} + 25\cdot 229^{3} + 121\cdot 229^{4} + 176\cdot 229^{5} +O\left(229^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 175 + 117\cdot 229 + 41\cdot 229^{2} + 129\cdot 229^{3} + 151\cdot 229^{4} + 177\cdot 229^{5} +O\left(229^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 188 + 155\cdot 229 + 17\cdot 229^{2} + 108\cdot 229^{3} + 69\cdot 229^{4} + 103\cdot 229^{5} +O\left(229^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 191 + 40\cdot 229 + 151\cdot 229^{2} + 53\cdot 229^{3} + 62\cdot 229^{4} + 30\cdot 229^{5} +O\left(229^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 192 + 38\cdot 229 + 143\cdot 229^{2} + 192\cdot 229^{3} + 94\cdot 229^{4} + 183\cdot 229^{5} +O\left(229^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 196 + 164\cdot 229 + 194\cdot 229^{2} + 107\cdot 229^{3} + 187\cdot 229^{4} + 184\cdot 229^{5} +O\left(229^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 212 + 160\cdot 229 + 28\cdot 229^{2} + 166\cdot 229^{3} + 106\cdot 229^{4} + 220\cdot 229^{5} +O\left(229^{ 6 }\right)$

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

Cycle notation

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

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

$(4,8)(5,6)$

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.