Artin representation 4.2e17_3e2.8t21.2c1

• Label: 4.2e17_3e2.8t21.2c1
• Dimension: 4
• Group: $C_2^3 : C_4 $
• Conductor: $ 2^{17} \cdot 3^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : C_4 $
Conductor:	$1179648= 2^{17} \cdot 3^{2} $
Artin number field:	Splitting field of $f= x^{8} - 4 x^{4} - 12 x^{2} + 25 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$C_2^3: C_4$
Parity:	Even
Determinant:	1.2e3.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 233 }$ to precision 7.

Roots:

$r_{ 1 }$	$=$	$ 7 + 87\cdot 233^{2} + 158\cdot 233^{3} + 98\cdot 233^{4} + 56\cdot 233^{5} + 134\cdot 233^{6} +O\left(233^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 15 + 115\cdot 233 + 170\cdot 233^{2} + 215\cdot 233^{3} + 231\cdot 233^{4} + 134\cdot 233^{5} + 195\cdot 233^{6} +O\left(233^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 23 + 82\cdot 233 + 26\cdot 233^{2} + 193\cdot 233^{3} + 190\cdot 233^{4} + 27\cdot 233^{5} + 32\cdot 233^{6} +O\left(233^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 96 + 109\cdot 233 + 184\cdot 233^{2} + 153\cdot 233^{3} + 54\cdot 233^{4} + 191\cdot 233^{5} + 136\cdot 233^{6} +O\left(233^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 137 + 123\cdot 233 + 48\cdot 233^{2} + 79\cdot 233^{3} + 178\cdot 233^{4} + 41\cdot 233^{5} + 96\cdot 233^{6} +O\left(233^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 210 + 150\cdot 233 + 206\cdot 233^{2} + 39\cdot 233^{3} + 42\cdot 233^{4} + 205\cdot 233^{5} + 200\cdot 233^{6} +O\left(233^{ 7 }\right)$
$r_{ 7 }$	$=$	$ 218 + 117\cdot 233 + 62\cdot 233^{2} + 17\cdot 233^{3} + 233^{4} + 98\cdot 233^{5} + 37\cdot 233^{6} +O\left(233^{ 7 }\right)$
$r_{ 8 }$	$=$	$ 226 + 232\cdot 233 + 145\cdot 233^{2} + 74\cdot 233^{3} + 134\cdot 233^{4} + 176\cdot 233^{5} + 98\cdot 233^{6} +O\left(233^{ 7 }\right)$

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

Cycle notation

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

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.