Artin representation 4.5e3_19e4.8t30.4c1

• Label: 4.5e3_19e4.8t30.4c1
• Dimension: 4
• Group: $(((C_4 \times C_2): C_2):C_2):C_2$
• Conductor: $ 5^{3} \cdot 19^{4}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$(((C_4 \times C_2): C_2):C_2):C_2$
Conductor:	$16290125= 5^{3} \cdot 19^{4} $
Artin number field:	Splitting field of $f= x^{8} - x^{7} + 2 x^{6} - 5 x^{5} + x^{4} - 5 x^{3} + 2 x^{2} - x + 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$(((C_4 \times C_2): C_2):C_2):C_2$
Parity:	Even
Determinant:	1.5.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 191 }$ to precision 7.

Roots:

$r_{ 1 }$	$=$	$ 28 + 124\cdot 191 + 162\cdot 191^{2} + 108\cdot 191^{3} + 109\cdot 191^{4} + 48\cdot 191^{5} + 57\cdot 191^{6} +O\left(191^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 31 + 68\cdot 191 + 155\cdot 191^{2} + 49\cdot 191^{3} + 106\cdot 191^{4} + 165\cdot 191^{5} + 53\cdot 191^{6} +O\left(191^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 37 + 85\cdot 191 + 22\cdot 191^{2} + 70\cdot 191^{3} + 144\cdot 191^{4} + 59\cdot 191^{5} + 143\cdot 191^{6} +O\left(191^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 53 + 142\cdot 191 + 4\cdot 191^{2} + 41\cdot 191^{3} + 191^{4} + 189\cdot 191^{5} + 105\cdot 191^{6} +O\left(191^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 116 + 161\cdot 191 + 38\cdot 191^{2} + 62\cdot 191^{3} + 164\cdot 191^{4} + 62\cdot 191^{5} + 90\cdot 191^{6} +O\left(191^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 150 + 138\cdot 191 + 145\cdot 191^{2} + 53\cdot 191^{3} + 26\cdot 191^{4} + 168\cdot 191^{5} + 127\cdot 191^{6} +O\left(191^{ 7 }\right)$
$r_{ 7 }$	$=$	$ 173 + 123\cdot 191 + 112\cdot 191^{2} + 113\cdot 191^{3} + 110\cdot 191^{4} + 8\cdot 191^{5} + 13\cdot 191^{6} +O\left(191^{ 7 }\right)$
$r_{ 8 }$	$=$	$ 177 + 110\cdot 191 + 121\cdot 191^{2} + 73\cdot 191^{3} + 101\cdot 191^{4} + 61\cdot 191^{5} + 172\cdot 191^{6} +O\left(191^{ 7 }\right)$

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

Cycle notation

$(2,3)(4,7)$

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

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

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

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

$(2,7,3,4)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.