Artin representation 4.2e8_5e3_11e2.8t29.2c1

• Label: 4.2e8_5e3_11e2.8t29.2c1
• Dimension: 4
• Group: $(((C_4 \times C_2): C_2):C_2):C_2$
• Conductor: $ 2^{8} \cdot 5^{3} \cdot 11^{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:	$3872000= 2^{8} \cdot 5^{3} \cdot 11^{2} $
Artin number field:	Splitting field of $f= x^{8} - 2 x^{7} + 6 x^{6} - 4 x^{5} + 6 x^{4} + 4 x^{3} + 4 x + 2 $ 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 $\Q_{ 509 }$ to precision 7.

Roots:

$r_{ 1 }$	$=$	$ 99 + 463\cdot 509 + 352\cdot 509^{2} + 93\cdot 509^{3} + 381\cdot 509^{4} + 21\cdot 509^{5} + 340\cdot 509^{6} +O\left(509^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 160 + 140\cdot 509 + 481\cdot 509^{2} + 362\cdot 509^{3} + 100\cdot 509^{4} + 252\cdot 509^{5} + 46\cdot 509^{6} +O\left(509^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 311 + 257\cdot 509 + 193\cdot 509^{2} + 99\cdot 509^{3} + 301\cdot 509^{4} + 340\cdot 509^{5} + 464\cdot 509^{6} +O\left(509^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 373 + 90\cdot 509 + 37\cdot 509^{2} + 91\cdot 509^{3} + 265\cdot 509^{4} + 508\cdot 509^{5} + 412\cdot 509^{6} +O\left(509^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 374 + 245\cdot 509 + 157\cdot 509^{2} + 231\cdot 509^{3} + 197\cdot 509^{4} + 489\cdot 509^{5} + 287\cdot 509^{6} +O\left(509^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 382 + 52\cdot 509 + 290\cdot 509^{2} + 213\cdot 509^{3} + 345\cdot 509^{4} + 284\cdot 509^{5} + 417\cdot 509^{6} +O\left(509^{ 7 }\right)$
$r_{ 7 }$	$=$	$ 417 + 335\cdot 509 + 498\cdot 509^{2} + 301\cdot 509^{3} + 279\cdot 509^{4} + 181\cdot 509^{5} + 293\cdot 509^{6} +O\left(509^{ 7 }\right)$
$r_{ 8 }$	$=$	$ 431 + 449\cdot 509 + 24\cdot 509^{2} + 133\cdot 509^{3} + 165\cdot 509^{4} + 466\cdot 509^{5} + 281\cdot 509^{6} +O\left(509^{ 7 }\right)$

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

Cycle notation

$(2,6)(4,7)$

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

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

$(1,8)(4,7)$

$(2,6)(3,5)$

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.