Galois orbit of Artin representations 4.5e4_11e2_31e2.8t16.6

• Label: 4.5e4_11e2_31e2.8t16.6
• Dimension: 4
• Group: $(C_8:C_2):C_2$
• Conductor: $ 5^{4} \cdot 11^{2} \cdot 31^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$(C_8:C_2):C_2$
Conductor:	$72675625= 5^{4} \cdot 11^{2} \cdot 31^{2} $
Artin number field:	Splitting field of $f= x^{8} - 3 x^{7} + 23 x^{6} - 31 x^{5} + 200 x^{4} - 176 x^{3} + 1073 x^{2} - 373 x + 2141 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$(C_8:C_2):C_2$
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 71 }$ to precision 7.

Roots:

$r_{ 1 }$	$=$	$ 6 + 17\cdot 71 + 27\cdot 71^{2} + 11\cdot 71^{3} + 11\cdot 71^{4} + 56\cdot 71^{5} + 27\cdot 71^{6} +O\left(71^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 19 + 28\cdot 71 + 25\cdot 71^{2} + 43\cdot 71^{3} + 62\cdot 71^{4} + 56\cdot 71^{5} + 23\cdot 71^{6} +O\left(71^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 24 + 6\cdot 71 + 49\cdot 71^{2} + 50\cdot 71^{3} + 66\cdot 71^{4} + 69\cdot 71^{5} + 27\cdot 71^{6} +O\left(71^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 34 + 12\cdot 71 + 26\cdot 71^{2} + 49\cdot 71^{3} + 67\cdot 71^{4} + 41\cdot 71^{5} + 23\cdot 71^{6} +O\left(71^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 40 + 20\cdot 71 + 61\cdot 71^{2} + 62\cdot 71^{3} + 26\cdot 71^{4} + 61\cdot 71^{5} + 66\cdot 71^{6} +O\left(71^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 49 + 21\cdot 71 + 71^{2} + 42\cdot 71^{3} + 43\cdot 71^{4} + 20\cdot 71^{5} + 10\cdot 71^{6} +O\left(71^{ 7 }\right)$
$r_{ 7 }$	$=$	$ 57 + 36\cdot 71 + 65\cdot 71^{2} + 32\cdot 71^{3} + 33\cdot 71^{4} + 9\cdot 71^{5} + 67\cdot 71^{6} +O\left(71^{ 7 }\right)$
$r_{ 8 }$	$=$	$ 58 + 69\cdot 71 + 27\cdot 71^{2} + 62\cdot 71^{3} + 42\cdot 71^{4} + 38\cdot 71^{5} + 36\cdot 71^{6} +O\left(71^{ 7 }\right)$

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

Cycle notation

$(3,4)(5,7)$

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

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

$(2,8)(5,7)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.