Artin representation 4.5e3_41e4.8t21.2c1

• Label: 4.5e3_41e4.8t21.2c1
• Dimension: 4
• Group: $C_2^3 : C_4 $
• Conductor: $ 5^{3} \cdot 41^{4}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : C_4 $
Conductor:	$353220125= 5^{3} \cdot 41^{4} $
Artin number field:	Splitting field of $f= x^{8} - 3 x^{7} - 8 x^{6} + 120 x^{5} - 219 x^{4} - 700 x^{3} + 1878 x^{2} + 161 x - 2049 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$C_2^3: C_4$
Parity:	Even
Determinant:	1.5.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 131 }$ to precision 7.

Roots:

$r_{ 1 }$	$=$	$ 40 + 127\cdot 131 + 2\cdot 131^{2} + 74\cdot 131^{3} + 100\cdot 131^{4} + 105\cdot 131^{6} +O\left(131^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 41 + 20\cdot 131 + 104\cdot 131^{2} + 114\cdot 131^{3} + 40\cdot 131^{4} + 13\cdot 131^{5} + 29\cdot 131^{6} +O\left(131^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 43 + 50\cdot 131 + 58\cdot 131^{2} + 123\cdot 131^{3} + 35\cdot 131^{4} + 84\cdot 131^{5} + 49\cdot 131^{6} +O\left(131^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 45 + 30\cdot 131 + 118\cdot 131^{2} + 128\cdot 131^{3} + 67\cdot 131^{4} + 55\cdot 131^{5} + 4\cdot 131^{6} +O\left(131^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 49 + 14\cdot 131 + 46\cdot 131^{2} + 64\cdot 131^{3} + 58\cdot 131^{4} + 39\cdot 131^{5} + 92\cdot 131^{6} +O\left(131^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 77 + 117\cdot 131 + 67\cdot 131^{2} + 8\cdot 131^{3} + 51\cdot 131^{4} + 41\cdot 131^{5} +O\left(131^{ 7 }\right)$
$r_{ 7 }$	$=$	$ 110 + 75\cdot 131 + 15\cdot 131^{2} + 81\cdot 131^{3} + 99\cdot 131^{4} + 30\cdot 131^{5} + 69\cdot 131^{6} +O\left(131^{ 7 }\right)$
$r_{ 8 }$	$=$	$ 122 + 87\cdot 131 + 110\cdot 131^{2} + 59\cdot 131^{3} + 69\cdot 131^{4} + 127\cdot 131^{5} + 42\cdot 131^{6} +O\left(131^{ 7 }\right)$

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

Cycle notation

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

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

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

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

$(1,8)(3,4)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.