Galois orbit of Artin representations 4.2e8_5e2_19e2.8t15.1

• Label: 4.2e8_5e2_19e2.8t15.1
• Dimension: 4
• Group: $Z_8 : Z_8^\times$
• Conductor: $ 2^{8} \cdot 5^{2} \cdot 19^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$Z_8 : Z_8^\times$
Conductor:	$2310400= 2^{8} \cdot 5^{2} \cdot 19^{2} $
Artin number field:	Splitting field of $f= x^{8} + 5 x^{6} + 3 x^{4} - 15 x^{2} - 19 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$Z_8 : Z_8^\times$
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 461 }$ to precision 8.

Roots:

$r_{ 1 }$	$=$	$ 7 + 225\cdot 461 + 367\cdot 461^{2} + 132\cdot 461^{3} + 37\cdot 461^{4} + 268\cdot 461^{5} + 412\cdot 461^{6} + 358\cdot 461^{7} +O\left(461^{ 8 }\right)$
$r_{ 2 }$	$=$	$ 71 + 327\cdot 461 + 412\cdot 461^{2} + 373\cdot 461^{3} + 18\cdot 461^{4} + 192\cdot 461^{5} + 177\cdot 461^{6} + 186\cdot 461^{7} +O\left(461^{ 8 }\right)$
$r_{ 3 }$	$=$	$ 176 + 195\cdot 461 + 17\cdot 461^{2} + 214\cdot 461^{3} + 434\cdot 461^{4} + 82\cdot 461^{5} + 422\cdot 461^{6} + 280\cdot 461^{7} +O\left(461^{ 8 }\right)$
$r_{ 4 }$	$=$	$ 229 + 287\cdot 461 + 174\cdot 461^{2} + 12\cdot 461^{3} + 447\cdot 461^{4} + 326\cdot 461^{5} + 366\cdot 461^{6} + 198\cdot 461^{7} +O\left(461^{ 8 }\right)$
$r_{ 5 }$	$=$	$ 232 + 173\cdot 461 + 286\cdot 461^{2} + 448\cdot 461^{3} + 13\cdot 461^{4} + 134\cdot 461^{5} + 94\cdot 461^{6} + 262\cdot 461^{7} +O\left(461^{ 8 }\right)$
$r_{ 6 }$	$=$	$ 285 + 265\cdot 461 + 443\cdot 461^{2} + 246\cdot 461^{3} + 26\cdot 461^{4} + 378\cdot 461^{5} + 38\cdot 461^{6} + 180\cdot 461^{7} +O\left(461^{ 8 }\right)$
$r_{ 7 }$	$=$	$ 390 + 133\cdot 461 + 48\cdot 461^{2} + 87\cdot 461^{3} + 442\cdot 461^{4} + 268\cdot 461^{5} + 283\cdot 461^{6} + 274\cdot 461^{7} +O\left(461^{ 8 }\right)$
$r_{ 8 }$	$=$	$ 454 + 235\cdot 461 + 93\cdot 461^{2} + 328\cdot 461^{3} + 423\cdot 461^{4} + 192\cdot 461^{5} + 48\cdot 461^{6} + 102\cdot 461^{7} +O\left(461^{ 8 }\right)$

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

Cycle notation

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

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

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

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

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

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 8 }$	Character values
			$c1$
$1$	$1$	$()$	$4$
$1$	$2$	$(1,8)(2,7)(3,6)(4,5)$	$-4$
$2$	$2$	$(1,8)(2,7)$	$0$
$4$	$2$	$(1,6)(2,4)(3,8)(5,7)$	$0$
$4$	$2$	$(1,7)(2,8)(4,5)$	$0$
$4$	$2$	$(1,8)(3,4)(5,6)$	$0$
$2$	$4$	$(1,7,8,2)(3,4,6,5)$	$0$
$2$	$4$	$(1,7,8,2)(3,5,6,4)$	$0$
$4$	$4$	$(1,5,8,4)(2,6,7,3)$	$0$
$4$	$8$	$(1,5,7,6,8,4,2,3)$	$0$
$4$	$8$	$(1,4,2,6,8,5,7,3)$	$0$

The blue line marks the conjugacy class containing complex conjugation.