Galois orbit of Artin representations 4.2e12_5e3.8t21.3

• Label: 4.2e12_5e3.8t21.3
• Dimension: 4
• Group: $C_2^3 : C_4 $
• Conductor: $ 2^{12} \cdot 5^{3}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : C_4 $
Conductor:	$512000= 2^{12} \cdot 5^{3} $
Artin number field:	Splitting field of $f= x^{8} - 4 x^{6} - 4 x^{5} + 24 x^{4} - 40 x^{3} + 40 x^{2} - 20 x + 5 $ 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 $\Q_{ 61 }$ to precision 8.

Roots:

$r_{ 1 }$	$=$	$ 2 + 31\cdot 61 + 51\cdot 61^{2} + 35\cdot 61^{3} + 43\cdot 61^{4} + 35\cdot 61^{5} + 48\cdot 61^{6} + 60\cdot 61^{7} +O\left(61^{ 8 }\right)$
$r_{ 2 }$	$=$	$ 16 + 32\cdot 61 + 7\cdot 61^{2} + 12\cdot 61^{3} + 14\cdot 61^{4} + 60\cdot 61^{5} + 19\cdot 61^{6} + 32\cdot 61^{7} +O\left(61^{ 8 }\right)$
$r_{ 3 }$	$=$	$ 19 + 32\cdot 61 + 45\cdot 61^{2} + 51\cdot 61^{3} + 48\cdot 61^{4} + 21\cdot 61^{5} + 24\cdot 61^{6} + 52\cdot 61^{7} +O\left(61^{ 8 }\right)$
$r_{ 4 }$	$=$	$ 24 + 26\cdot 61 + 17\cdot 61^{2} + 22\cdot 61^{3} + 15\cdot 61^{4} + 4\cdot 61^{5} + 29\cdot 61^{6} + 37\cdot 61^{7} +O\left(61^{ 8 }\right)$
$r_{ 5 }$	$=$	$ 32 + 60\cdot 61 + 35\cdot 61^{2} + 15\cdot 61^{3} + 23\cdot 61^{4} + 20\cdot 61^{5} + 60\cdot 61^{6} + 41\cdot 61^{7} +O\left(61^{ 8 }\right)$
$r_{ 6 }$	$=$	$ 36 + 37\cdot 61 + 23\cdot 61^{2} + 38\cdot 61^{3} + 32\cdot 61^{4} + 49\cdot 61^{5} + 8\cdot 61^{6} + 29\cdot 61^{7} +O\left(61^{ 8 }\right)$
$r_{ 7 }$	$=$	$ 56 + 52\cdot 61 + 26\cdot 61^{2} + 36\cdot 61^{3} + 48\cdot 61^{4} + 22\cdot 61^{5} + 14\cdot 61^{6} + 28\cdot 61^{7} +O\left(61^{ 8 }\right)$
$r_{ 8 }$	$=$	$ 59 + 31\cdot 61 + 35\cdot 61^{2} + 31\cdot 61^{3} + 17\cdot 61^{4} + 29\cdot 61^{5} + 38\cdot 61^{6} + 22\cdot 61^{7} +O\left(61^{ 8 }\right)$

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

Cycle notation

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

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.