Artin representation 4.2e12_3e4.8t15.1c1

• Label: 4.2e12_3e4.8t15.1c1
• Dimension: 4
• Group: $Z_8 : Z_8^\times$
• Conductor: $ 2^{12} \cdot 3^{4}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$Z_8 : Z_8^\times$
Conductor:	$331776= 2^{12} \cdot 3^{4} $
Artin number field:	Splitting field of $f= x^{8} - 6 x^{6} + 15 x^{4} - 12 x^{2} + 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 $\Q_{ 433 }$ to precision 8.

Roots:

$r_{ 1 }$	$=$	$ 29 + 391\cdot 433 + 233\cdot 433^{2} + 337\cdot 433^{3} + 66\cdot 433^{4} + 168\cdot 433^{5} + 115\cdot 433^{6} + 83\cdot 433^{7} +O\left(433^{ 8 }\right)$
$r_{ 2 }$	$=$	$ 101 + 195\cdot 433 + 200\cdot 433^{2} + 54\cdot 433^{3} + 155\cdot 433^{4} + 96\cdot 433^{5} + 353\cdot 433^{6} + 335\cdot 433^{7} +O\left(433^{ 8 }\right)$
$r_{ 3 }$	$=$	$ 138 + 109\cdot 433 + 269\cdot 433^{2} + 91\cdot 433^{3} + 42\cdot 433^{4} + 126\cdot 433^{5} + 238\cdot 433^{6} + 389\cdot 433^{7} +O\left(433^{ 8 }\right)$
$r_{ 4 }$	$=$	$ 171 + 53\cdot 433 + 392\cdot 433^{2} + 410\cdot 433^{3} + 29\cdot 433^{4} + 31\cdot 433^{5} + 252\cdot 433^{6} + 63\cdot 433^{7} +O\left(433^{ 8 }\right)$
$r_{ 5 }$	$=$	$ 262 + 379\cdot 433 + 40\cdot 433^{2} + 22\cdot 433^{3} + 403\cdot 433^{4} + 401\cdot 433^{5} + 180\cdot 433^{6} + 369\cdot 433^{7} +O\left(433^{ 8 }\right)$
$r_{ 6 }$	$=$	$ 295 + 323\cdot 433 + 163\cdot 433^{2} + 341\cdot 433^{3} + 390\cdot 433^{4} + 306\cdot 433^{5} + 194\cdot 433^{6} + 43\cdot 433^{7} +O\left(433^{ 8 }\right)$
$r_{ 7 }$	$=$	$ 332 + 237\cdot 433 + 232\cdot 433^{2} + 378\cdot 433^{3} + 277\cdot 433^{4} + 336\cdot 433^{5} + 79\cdot 433^{6} + 97\cdot 433^{7} +O\left(433^{ 8 }\right)$
$r_{ 8 }$	$=$	$ 404 + 41\cdot 433 + 199\cdot 433^{2} + 95\cdot 433^{3} + 366\cdot 433^{4} + 264\cdot 433^{5} + 317\cdot 433^{6} + 349\cdot 433^{7} +O\left(433^{ 8 }\right)$

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

Cycle notation

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

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.