Artin representation 4.5e2_11_29.8t35.4c1

• Label: 4.5e2_11_29.8t35.4c1
• Dimension: 4
• Group: $C_2 \wr C_2\wr C_2$
• Conductor: $ 5^{2} \cdot 11 \cdot 29 $
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2 \wr C_2\wr C_2$
Conductor:	$7975= 5^{2} \cdot 11 \cdot 29 $
Artin number field:	Splitting field of $f= x^{8} - 2 x^{7} + 4 x^{6} - 5 x^{5} + 5 x^{4} - 5 x^{3} + 4 x^{2} - 2 x + 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$C_2 \wr C_2\wr C_2$
Parity:	Odd
Determinant:	1.11_29.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 929 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 87 + 443\cdot 929 + 885\cdot 929^{2} + 341\cdot 929^{3} + 239\cdot 929^{4} + 712\cdot 929^{5} +O\left(929^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 97 + 391\cdot 929 + 161\cdot 929^{2} + 803\cdot 929^{3} + 490\cdot 929^{4} + 815\cdot 929^{5} +O\left(929^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 233 + 383\cdot 929 + 561\cdot 929^{2} + 307\cdot 929^{3} + 682\cdot 929^{4} + 669\cdot 929^{5} +O\left(929^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 299 + 474\cdot 929 + 74\cdot 929^{2} + 212\cdot 929^{3} + 372\cdot 929^{4} + 757\cdot 929^{5} +O\left(929^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 311 + 557\cdot 929 + 336\cdot 929^{2} + 67\cdot 929^{3} + 564\cdot 929^{4} + 647\cdot 929^{5} +O\left(929^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 358 + 377\cdot 929 + 73\cdot 929^{2} + 545\cdot 929^{3} + 634\cdot 929^{4} + 524\cdot 929^{5} +O\left(929^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 680 + 821\cdot 929 + 251\cdot 929^{2} + 697\cdot 929^{3} + 420\cdot 929^{4} + 482\cdot 929^{5} +O\left(929^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 724 + 267\cdot 929 + 442\cdot 929^{2} + 741\cdot 929^{3} + 311\cdot 929^{4} + 35\cdot 929^{5} +O\left(929^{ 6 }\right)$

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

Cycle notation

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

$(1,4)$

$(6,8)$

$(2,7)$

$(1,3)(4,5)$

$(3,5)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.