Artin representation 4.5e3_41e2.8t30.3c1

• Label: 4.5e3_41e2.8t30.3c1
• Dimension: 4
• Group: $(((C_4 \times C_2): C_2):C_2):C_2$
• Conductor: $ 5^{3} \cdot 41^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$(((C_4 \times C_2): C_2):C_2):C_2$
Conductor:	$210125= 5^{3} \cdot 41^{2} $
Artin number field:	Splitting field of $f= x^{8} - 3 x^{7} + 2 x^{6} + 4 x^{5} - x^{4} - 4 x^{3} + 2 x^{2} + 3 x + 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$(((C_4 \times C_2): C_2):C_2):C_2$
Parity:	Even
Determinant:	1.5.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 541 }$ to precision 7.

Roots:

$r_{ 1 }$	$=$	$ 110 + 478\cdot 541 + 11\cdot 541^{2} + 164\cdot 541^{3} + 285\cdot 541^{4} + 199\cdot 541^{5} + 115\cdot 541^{6} +O\left(541^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 189 + 185\cdot 541 + 444\cdot 541^{2} + 533\cdot 541^{3} + 244\cdot 541^{4} + 207\cdot 541^{5} + 95\cdot 541^{6} +O\left(541^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 290 + 433\cdot 541 + 427\cdot 541^{2} + 218\cdot 541^{3} + 337\cdot 541^{4} + 332\cdot 541^{5} + 119\cdot 541^{6} +O\left(541^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 300 + 127\cdot 541 + 286\cdot 541^{2} + 76\cdot 541^{3} + 68\cdot 541^{4} + 329\cdot 541^{5} + 64\cdot 541^{6} +O\left(541^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 308 + 501\cdot 541 + 418\cdot 541^{2} + 492\cdot 541^{3} + 265\cdot 541^{4} + 54\cdot 541^{5} + 86\cdot 541^{6} +O\left(541^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 312 + 298\cdot 541 + 82\cdot 541^{2} + 18\cdot 541^{3} + 384\cdot 541^{4} + 530\cdot 541^{5} + 74\cdot 541^{6} +O\left(541^{ 7 }\right)$
$r_{ 7 }$	$=$	$ 319 + 135\cdot 541 + 85\cdot 541^{2} + 45\cdot 541^{3} + 213\cdot 541^{4} + 181\cdot 541^{5} + 410\cdot 541^{6} +O\left(541^{ 7 }\right)$
$r_{ 8 }$	$=$	$ 339 + 3\cdot 541 + 407\cdot 541^{2} + 73\cdot 541^{3} + 365\cdot 541^{4} + 328\cdot 541^{5} + 115\cdot 541^{6} +O\left(541^{ 7 }\right)$

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

Cycle notation

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

$(2,6)(5,8)$

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

$(3,7)(5,8)$

$(1,4)(2,6)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.