Artin representation 4.2e12_3e2_7e2.8t15.5c1

• Label: 4.2e12_3e2_7e2.8t15.5c1
• Dimension: 4
• Group: $Z_8 : Z_8^\times$
• Conductor: $ 2^{12} \cdot 3^{2} \cdot 7^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$Z_8 : Z_8^\times$
Conductor:	$1806336= 2^{12} \cdot 3^{2} \cdot 7^{2} $
Artin number field:	Splitting field of $f= x^{8} - 2 x^{6} - x^{4} + 8 x^{2} + 7 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$Z_8 : Z_8^\times$
Parity:	Even
Determinant:	1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 199 }$ to precision 8.

Roots:

$r_{ 1 }$	$=$	$ 11 + 48\cdot 199 + 186\cdot 199^{2} + 79\cdot 199^{3} + 168\cdot 199^{4} + 47\cdot 199^{5} + 173\cdot 199^{6} + 106\cdot 199^{7} +O\left(199^{ 8 }\right)$
$r_{ 2 }$	$=$	$ 57 + 150\cdot 199 + 24\cdot 199^{2} + 35\cdot 199^{3} + 121\cdot 199^{4} + 177\cdot 199^{5} + 14\cdot 199^{6} + 156\cdot 199^{7} +O\left(199^{ 8 }\right)$
$r_{ 3 }$	$=$	$ 67 + 177\cdot 199 + 187\cdot 199^{2} + 14\cdot 199^{3} + 141\cdot 199^{4} + 62\cdot 199^{5} + 43\cdot 199^{6} + 124\cdot 199^{7} +O\left(199^{ 8 }\right)$
$r_{ 4 }$	$=$	$ 74 + 81\cdot 199 + 22\cdot 199^{2} + 145\cdot 199^{3} + 190\cdot 199^{4} + 142\cdot 199^{5} + 61\cdot 199^{6} + 10\cdot 199^{7} +O\left(199^{ 8 }\right)$
$r_{ 5 }$	$=$	$ 125 + 117\cdot 199 + 176\cdot 199^{2} + 53\cdot 199^{3} + 8\cdot 199^{4} + 56\cdot 199^{5} + 137\cdot 199^{6} + 188\cdot 199^{7} +O\left(199^{ 8 }\right)$
$r_{ 6 }$	$=$	$ 132 + 21\cdot 199 + 11\cdot 199^{2} + 184\cdot 199^{3} + 57\cdot 199^{4} + 136\cdot 199^{5} + 155\cdot 199^{6} + 74\cdot 199^{7} +O\left(199^{ 8 }\right)$
$r_{ 7 }$	$=$	$ 142 + 48\cdot 199 + 174\cdot 199^{2} + 163\cdot 199^{3} + 77\cdot 199^{4} + 21\cdot 199^{5} + 184\cdot 199^{6} + 42\cdot 199^{7} +O\left(199^{ 8 }\right)$
$r_{ 8 }$	$=$	$ 188 + 150\cdot 199 + 12\cdot 199^{2} + 119\cdot 199^{3} + 30\cdot 199^{4} + 151\cdot 199^{5} + 25\cdot 199^{6} + 92\cdot 199^{7} +O\left(199^{ 8 }\right)$

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

Cycle notation

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

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.