Galois orbit of Artin representations 4.2e4_3e2_11e2_13e2.8t22.5

• Label: 4.2e4_3e2_11e2_13e2.8t22.5
• Dimension: 4
• Group: $C_2^3 : D_4 $
• Conductor: $ 2^{4} \cdot 3^{2} \cdot 11^{2} \cdot 13^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : D_4 $
Conductor:	$2944656= 2^{4} \cdot 3^{2} \cdot 11^{2} \cdot 13^{2} $
Artin number field:	Splitting field of $f= x^{8} - 2 x^{7} - 5 x^{6} + 10 x^{5} + 15 x^{4} - 46 x^{3} - 17 x^{2} + 68 x + 73 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$C_2^3 : D_4 $
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 313 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 66 + 305\cdot 313 + 152\cdot 313^{2} + 208\cdot 313^{3} + 146\cdot 313^{4} + 211\cdot 313^{5} +O\left(313^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 97 + 311\cdot 313 + 247\cdot 313^{2} + 234\cdot 313^{3} + 183\cdot 313^{4} + 160\cdot 313^{5} +O\left(313^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 118 + 52\cdot 313 + 283\cdot 313^{2} + 79\cdot 313^{3} + 74\cdot 313^{4} + 25\cdot 313^{5} +O\left(313^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 149 + 58\cdot 313 + 65\cdot 313^{2} + 106\cdot 313^{3} + 111\cdot 313^{4} + 287\cdot 313^{5} +O\left(313^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 173 + 140\cdot 313 + 71\cdot 313^{2} + 268\cdot 313^{3} + 215\cdot 313^{4} + 146\cdot 313^{5} +O\left(313^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 183 + 223\cdot 313 + 257\cdot 313^{2} + 219\cdot 313^{3} + 188\cdot 313^{4} + 161\cdot 313^{5} +O\left(313^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 229 + 38\cdot 313 + 150\cdot 313^{2} + 91\cdot 313^{3} + 179\cdot 313^{4} + 278\cdot 313^{5} +O\left(313^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 239 + 121\cdot 313 + 23\cdot 313^{2} + 43\cdot 313^{3} + 152\cdot 313^{4} + 293\cdot 313^{5} +O\left(313^{ 6 }\right)$

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

Cycle notation

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

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

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

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

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

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

$(1,4)(2,3)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.