Artin representation 4.2e12_3e2_7e2.8t15.3c1

• Label: 4.2e12_3e2_7e2.8t15.3c1
• 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} - 8 x^{6} + 20 x^{4} - 16 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 $\Q_{ 337 }$ to precision 8.

Roots:

$r_{ 1 }$	$=$	$ 39 + 167\cdot 337 + 300\cdot 337^{2} + 155\cdot 337^{3} + 102\cdot 337^{4} + 263\cdot 337^{5} + 321\cdot 337^{6} + 330\cdot 337^{7} +O\left(337^{ 8 }\right)$
$r_{ 2 }$	$=$	$ 93 + 125\cdot 337 + 309\cdot 337^{2} + 203\cdot 337^{3} + 206\cdot 337^{4} + 112\cdot 337^{5} + 143\cdot 337^{6} + 166\cdot 337^{7} +O\left(337^{ 8 }\right)$
$r_{ 3 }$	$=$	$ 96 + 256\cdot 337 + 179\cdot 337^{2} + 104\cdot 337^{3} + 76\cdot 337^{4} + 296\cdot 337^{5} + 308\cdot 337^{6} + 176\cdot 337^{7} +O\left(337^{ 8 }\right)$
$r_{ 4 }$	$=$	$ 98 + 193\cdot 337 + 292\cdot 337^{2} + 258\cdot 337^{3} + 285\cdot 337^{4} + 93\cdot 337^{5} + 73\cdot 337^{6} + 196\cdot 337^{7} +O\left(337^{ 8 }\right)$
$r_{ 5 }$	$=$	$ 239 + 143\cdot 337 + 44\cdot 337^{2} + 78\cdot 337^{3} + 51\cdot 337^{4} + 243\cdot 337^{5} + 263\cdot 337^{6} + 140\cdot 337^{7} +O\left(337^{ 8 }\right)$
$r_{ 6 }$	$=$	$ 241 + 80\cdot 337 + 157\cdot 337^{2} + 232\cdot 337^{3} + 260\cdot 337^{4} + 40\cdot 337^{5} + 28\cdot 337^{6} + 160\cdot 337^{7} +O\left(337^{ 8 }\right)$
$r_{ 7 }$	$=$	$ 244 + 211\cdot 337 + 27\cdot 337^{2} + 133\cdot 337^{3} + 130\cdot 337^{4} + 224\cdot 337^{5} + 193\cdot 337^{6} + 170\cdot 337^{7} +O\left(337^{ 8 }\right)$
$r_{ 8 }$	$=$	$ 298 + 169\cdot 337 + 36\cdot 337^{2} + 181\cdot 337^{3} + 234\cdot 337^{4} + 73\cdot 337^{5} + 15\cdot 337^{6} + 6\cdot 337^{7} +O\left(337^{ 8 }\right)$

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

Cycle notation

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

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.