Artin representation 4.2e3_7e2_37e3.8t35.1c1

• Label: 4.2e3_7e2_37e3.8t35.1c1
• Dimension: 4
• Group: $C_2 \wr C_2\wr C_2$
• Conductor: $ 2^{3} \cdot 7^{2} \cdot 37^{3}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2 \wr C_2\wr C_2$
Conductor:	$19855976= 2^{3} \cdot 7^{2} \cdot 37^{3} $
Artin number field:	Splitting field of $f= x^{8} - x^{7} + 2 x^{5} - x^{4} - 2 x^{3} + x + 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$C_2 \wr C_2\wr C_2$
Parity:	Even
Determinant:	1.2e3_37.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 743 }$ to precision 7.

Roots:

$r_{ 1 }$	$=$	$ 41 + 117\cdot 743 + 318\cdot 743^{2} + 43\cdot 743^{3} + 73\cdot 743^{4} + 513\cdot 743^{5} + 31\cdot 743^{6} +O\left(743^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 62 + 234\cdot 743 + 523\cdot 743^{2} + 298\cdot 743^{3} + 510\cdot 743^{4} + 579\cdot 743^{5} + 651\cdot 743^{6} +O\left(743^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 115 + 357\cdot 743 + 451\cdot 743^{2} + 565\cdot 743^{3} + 303\cdot 743^{4} + 586\cdot 743^{5} + 272\cdot 743^{6} +O\left(743^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 357 + 381\cdot 743 + 621\cdot 743^{2} + 55\cdot 743^{3} + 138\cdot 743^{4} + 629\cdot 743^{5} + 493\cdot 743^{6} +O\left(743^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 410 + 26\cdot 743 + 349\cdot 743^{2} + 602\cdot 743^{3} + 726\cdot 743^{4} + 465\cdot 743^{5} + 111\cdot 743^{6} +O\left(743^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 598 + 268\cdot 743 + 31\cdot 743^{2} + 406\cdot 743^{3} + 96\cdot 743^{4} + 578\cdot 743^{5} + 371\cdot 743^{6} +O\left(743^{ 7 }\right)$
$r_{ 7 }$	$=$	$ 659 + 570\cdot 743 + 443\cdot 743^{2} + 530\cdot 743^{3} + 24\cdot 743^{4} + 530\cdot 743^{5} + 441\cdot 743^{6} +O\left(743^{ 7 }\right)$
$r_{ 8 }$	$=$	$ 731 + 272\cdot 743 + 233\cdot 743^{2} + 469\cdot 743^{3} + 355\cdot 743^{4} + 575\cdot 743^{5} + 596\cdot 743^{6} +O\left(743^{ 7 }\right)$

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

Cycle notation

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

$(1,6)$

$(2,8)$

$(1,8)(2,6)$

$(4,5)$

$(3,7)$

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 8 }$	Character value
$1$	$1$	$()$	$4$
$1$	$2$	$(1,6)(2,8)(3,7)(4,5)$	$-4$
$2$	$2$	$(3,7)(4,5)$	$0$
$4$	$2$	$(3,7)$	$-2$
$4$	$2$	$(2,8)(3,7)$	$0$
$4$	$2$	$(1,8)(2,6)(3,5)(4,7)$	$0$
$4$	$2$	$(3,5)(4,7)$	$2$
$4$	$2$	$(1,6)(3,7)(4,5)$	$2$
$4$	$2$	$(1,8)(2,6)(3,7)(4,5)$	$-2$
$8$	$2$	$(1,4)(2,3)(5,6)(7,8)$	$0$
$8$	$2$	$(1,6)(3,5)(4,7)$	$0$
$4$	$4$	$(1,8,6,2)(3,4,7,5)$	$0$
$4$	$4$	$(3,4,7,5)$	$-2$
$4$	$4$	$(1,2,6,8)(3,7)(4,5)$	$2$
$8$	$4$	$(1,4,6,5)(2,3,8,7)$	$0$
$8$	$4$	$(1,6)(3,4,7,5)$	$0$
$8$	$4$	$(1,8)(2,6)(3,4,7,5)$	$0$
$16$	$4$	$(1,4)(2,3,8,7)(5,6)$	$0$
$16$	$4$	$(1,4,8,7)(2,3,6,5)$	$0$
$16$	$8$	$(1,4,8,7,6,5,2,3)$	$0$

The blue line marks the conjugacy class containing complex conjugation.