Artin representation 4.2e4_3e2_5e2_19e2.8t22.5c1

• Label: 4.2e4_3e2_5e2_19e2.8t22.5c1
• Dimension: 4
• Group: $C_2^3 : D_4 $
• Conductor: $ 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 19^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : D_4 $
Conductor:	$1299600= 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 19^{2} $
Artin number field:	Splitting field of $f= x^{8} - 3 x^{7} + 14 x^{6} - 32 x^{5} + 69 x^{4} - 92 x^{3} + 102 x^{2} - 65 x + 31 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$C_2^3 : D_4 $
Parity:	Even
Determinant:	1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 7 + 53\cdot 61 + 48\cdot 61^{2} + 4\cdot 61^{3} + 48\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 12 + 20\cdot 61 + 59\cdot 61^{2} + 57\cdot 61^{3} + 9\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 33 + 22\cdot 61 + 10\cdot 61^{2} + 58\cdot 61^{3} + 31\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 44 + 13\cdot 61 + 12\cdot 61^{2} + 29\cdot 61^{3} + 52\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 45 + 42\cdot 61 + 60\cdot 61^{2} + 30\cdot 61^{3} + 49\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 52 + 30\cdot 61 + 22\cdot 61^{2} + 55\cdot 61^{3} + 26\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 57 + 25\cdot 61 + 15\cdot 61^{2} + 5\cdot 61^{3} + 39\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 58 + 34\cdot 61 + 14\cdot 61^{2} + 2\cdot 61^{3} + 47\cdot 61^{4} +O\left(61^{ 5 }\right)$

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

Cycle notation

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

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

$(2,4)(7,8)$

$(1,5)(2,4)$

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

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

$(2,4)(3,6)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.