Artin representation 4.5e3_11e2_29e2.8t29.4c1

• Label: 4.5e3_11e2_29e2.8t29.4c1
• Dimension: 4
• Group: $(((C_4 \times C_2): C_2):C_2):C_2$
• Conductor: $ 5^{3} \cdot 11^{2} \cdot 29^{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:	$12720125= 5^{3} \cdot 11^{2} \cdot 29^{2} $
Artin number field:	Splitting field of $f= x^{8} - 3 x^{7} + 6 x^{6} - 11 x^{5} + x^{4} - 23 x^{3} + 44 x^{2} + 49 x + 71 $ 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 $\Q_{ 929 }$ to precision 8.

Roots:

$r_{ 1 }$	$=$	$ 147 + 421\cdot 929 + 745\cdot 929^{2} + 363\cdot 929^{3} + 775\cdot 929^{4} + 578\cdot 929^{5} + 889\cdot 929^{6} + 605\cdot 929^{7} +O\left(929^{ 8 }\right)$
$r_{ 2 }$	$=$	$ 232 + 875\cdot 929 + 636\cdot 929^{2} + 831\cdot 929^{3} + 301\cdot 929^{4} + 887\cdot 929^{5} + 404\cdot 929^{6} + 414\cdot 929^{7} +O\left(929^{ 8 }\right)$
$r_{ 3 }$	$=$	$ 308 + 603\cdot 929 + 833\cdot 929^{2} + 312\cdot 929^{3} + 402\cdot 929^{4} + 207\cdot 929^{5} + 188\cdot 929^{6} + 816\cdot 929^{7} +O\left(929^{ 8 }\right)$
$r_{ 4 }$	$=$	$ 312 + 347\cdot 929 + 439\cdot 929^{2} + 709\cdot 929^{3} + 748\cdot 929^{4} + 789\cdot 929^{5} + 211\cdot 929^{6} + 433\cdot 929^{7} +O\left(929^{ 8 }\right)$
$r_{ 5 }$	$=$	$ 439 + 628\cdot 929 + 706\cdot 929^{2} + 896\cdot 929^{3} + 842\cdot 929^{4} + 425\cdot 929^{5} + 207\cdot 929^{6} + 749\cdot 929^{7} +O\left(929^{ 8 }\right)$
$r_{ 6 }$	$=$	$ 625 + 202\cdot 929 + 67\cdot 929^{2} + 507\cdot 929^{3} + 643\cdot 929^{4} + 142\cdot 929^{5} + 499\cdot 929^{6} + 717\cdot 929^{7} +O\left(929^{ 8 }\right)$
$r_{ 7 }$	$=$	$ 767 + 420\cdot 929 + 104\cdot 929^{2} + 686\cdot 929^{3} + 80\cdot 929^{4} + 349\cdot 929^{5} + 694\cdot 929^{6} + 846\cdot 929^{7} +O\left(929^{ 8 }\right)$
$r_{ 8 }$	$=$	$ 889 + 216\cdot 929 + 182\cdot 929^{2} + 337\cdot 929^{3} + 849\cdot 929^{4} + 334\cdot 929^{5} + 620\cdot 929^{6} + 61\cdot 929^{7} +O\left(929^{ 8 }\right)$

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

Cycle notation

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

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

$(1,2)(5,8)$

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.