Artin representation 4.2e4_3e2_7e4.8t15.2c1

• Label: 4.2e4_3e2_7e4.8t15.2c1
• Dimension: 4
• Group: $Z_8 : Z_8^\times$
• Conductor: $ 2^{4} \cdot 3^{2} \cdot 7^{4}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$Z_8 : Z_8^\times$
Conductor:	$345744= 2^{4} \cdot 3^{2} \cdot 7^{4} $
Artin number field:	Splitting field of $f= x^{8} - 2 x^{7} + 7 x^{6} - 14 x^{5} + 14 x^{4} - 14 x^{2} + 12 x - 3 $ 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 $\Q_{ 193 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 45 + 50\cdot 193 + 138\cdot 193^{2} + 92\cdot 193^{3} + 36\cdot 193^{4} + 159\cdot 193^{5} +O\left(193^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 94 + 82\cdot 193 + 57\cdot 193^{2} + 64\cdot 193^{3} + 79\cdot 193^{4} + 8\cdot 193^{5} +O\left(193^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 111 + 166\cdot 193 + 175\cdot 193^{2} + 156\cdot 193^{3} + 97\cdot 193^{4} + 149\cdot 193^{5} +O\left(193^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 115 + 77\cdot 193 + 13\cdot 193^{2} + 139\cdot 193^{3} + 186\cdot 193^{4} + 145\cdot 193^{5} +O\left(193^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 135 + 189\cdot 193 + 35\cdot 193^{2} + 178\cdot 193^{3} + 13\cdot 193^{4} + 123\cdot 193^{5} +O\left(193^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 137 + 86\cdot 193 + 14\cdot 193^{2} + 72\cdot 193^{3} + 172\cdot 193^{4} + 68\cdot 193^{5} +O\left(193^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 158 + 37\cdot 193 + 182\cdot 193^{2} + 58\cdot 193^{3} + 57\cdot 193^{4} + 97\cdot 193^{5} +O\left(193^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 172 + 80\cdot 193 + 154\cdot 193^{2} + 9\cdot 193^{3} + 128\cdot 193^{4} + 19\cdot 193^{5} +O\left(193^{ 6 }\right)$

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

Cycle notation

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

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.