Artin representation 4.3e3_13e2_109e2.8t31.1c1

• Label: 4.3e3_13e2_109e2.8t31.1c1
• Dimension: 4
• Group: $(((C_4 \times C_2): C_2):C_2):C_2$
• Conductor: $ 3^{3} \cdot 13^{2} \cdot 109^{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:	$54213003= 3^{3} \cdot 13^{2} \cdot 109^{2} $
Artin number field:	Splitting field of $f= x^{8} - 3 x^{7} + 17 x^{6} - 35 x^{5} + 100 x^{4} - 89 x^{3} + 313 x^{2} + 63 x + 321 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$(((C_4 \times C_2): C_2):C_2):C_2$
Parity:	Odd
Determinant:	1.3.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 823 }$ to precision 8.

Roots:

$r_{ 1 }$	$=$	$ 106 + 663\cdot 823 + 318\cdot 823^{2} + 37\cdot 823^{3} + 138\cdot 823^{4} + 195\cdot 823^{5} + 812\cdot 823^{6} + 723\cdot 823^{7} +O\left(823^{ 8 }\right)$
$r_{ 2 }$	$=$	$ 167 + 751\cdot 823 + 806\cdot 823^{2} + 587\cdot 823^{3} + 601\cdot 823^{4} + 332\cdot 823^{5} + 745\cdot 823^{6} + 594\cdot 823^{7} +O\left(823^{ 8 }\right)$
$r_{ 3 }$	$=$	$ 362 + 234\cdot 823 + 765\cdot 823^{2} + 174\cdot 823^{3} + 314\cdot 823^{4} + 112\cdot 823^{5} + 716\cdot 823^{6} + 173\cdot 823^{7} +O\left(823^{ 8 }\right)$
$r_{ 4 }$	$=$	$ 631 + 162\cdot 823 + 768\cdot 823^{2} + 293\cdot 823^{3} + 7\cdot 823^{4} + 363\cdot 823^{5} + 234\cdot 823^{6} + 793\cdot 823^{7} +O\left(823^{ 8 }\right)$
$r_{ 5 }$	$=$	$ 651 + 308\cdot 823 + 234\cdot 823^{2} + 349\cdot 823^{3} + 657\cdot 823^{4} + 155\cdot 823^{5} + 504\cdot 823^{6} + 498\cdot 823^{7} +O\left(823^{ 8 }\right)$
$r_{ 6 }$	$=$	$ 672 + 742\cdot 823 + 690\cdot 823^{2} + 222\cdot 823^{3} + 477\cdot 823^{4} + 459\cdot 823^{5} + 10\cdot 823^{6} + 421\cdot 823^{7} +O\left(823^{ 8 }\right)$
$r_{ 7 }$	$=$	$ 751 + 655\cdot 823 + 752\cdot 823^{2} + 184\cdot 823^{3} + 211\cdot 823^{4} + 305\cdot 823^{5} + 59\cdot 823^{6} + 163\cdot 823^{7} +O\left(823^{ 8 }\right)$
$r_{ 8 }$	$=$	$ 778 + 595\cdot 823 + 600\cdot 823^{2} + 617\cdot 823^{3} + 61\cdot 823^{4} + 545\cdot 823^{5} + 209\cdot 823^{6} + 746\cdot 823^{7} +O\left(823^{ 8 }\right)$

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

Cycle notation

$(4,6)(7,8)$

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.