Artin representation 4.2e4_3e2_5e2_19e2.8t22.6c1

• Label: 4.2e4_3e2_5e2_19e2.8t22.6c1
• 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} - 2 x^{7} + 7 x^{6} - 18 x^{5} + 34 x^{4} - 64 x^{3} + 78 x^{2} - 84 x + 61 $ 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 6.

Roots:

$r_{ 1 }$	$=$	$ 8\cdot 61 + 15\cdot 61^{2} + 39\cdot 61^{3} + 46\cdot 61^{4} + 22\cdot 61^{5} +O\left(61^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 3 + 8\cdot 61 + 19\cdot 61^{2} + 21\cdot 61^{3} + 38\cdot 61^{4} + 28\cdot 61^{5} +O\left(61^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 10 + 15\cdot 61 + 38\cdot 61^{2} + 47\cdot 61^{3} + 48\cdot 61^{4} + 22\cdot 61^{5} +O\left(61^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 15 + 23\cdot 61 + 29\cdot 61^{2} + 7\cdot 61^{3} + 57\cdot 61^{4} + 50\cdot 61^{5} +O\left(61^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 16 + 26\cdot 61 + 55\cdot 61^{2} + 38\cdot 61^{3} + 8\cdot 61^{4} + 11\cdot 61^{5} +O\left(61^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 43 + 18\cdot 61 + 32\cdot 61^{2} + 22\cdot 61^{3} + 28\cdot 61^{4} + 59\cdot 61^{5} +O\left(61^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 47 + 16\cdot 61 + 29\cdot 61^{2} + 19\cdot 61^{3} + 61^{4} + 17\cdot 61^{5} +O\left(61^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 51 + 5\cdot 61 + 25\cdot 61^{2} + 47\cdot 61^{3} + 14\cdot 61^{4} + 31\cdot 61^{5} +O\left(61^{ 6 }\right)$

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

Cycle notation

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

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

$(3,4)(7,8)$

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

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

$(1,2)(7,8)$

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.