Artin representation 4.2e12_3e2_11e2.8t22.5c1

• Label: 4.2e12_3e2_11e2.8t22.5c1
• Dimension: 4
• Group: $C_2^3 : D_4 $
• Conductor: $ 2^{12} \cdot 3^{2} \cdot 11^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : D_4 $
Conductor:	$4460544= 2^{12} \cdot 3^{2} \cdot 11^{2} $
Artin number field:	Splitting field of $f= x^{8} - 10 x^{6} + 50 x^{4} - 110 x^{2} + 121 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$C_2^3 : D_4 $
Parity:	Even
Determinant:	1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 97 }$ to precision 7.

Roots:

$r_{ 1 }$	$=$	$ 2 + 18\cdot 97 + 70\cdot 97^{2} + 57\cdot 97^{3} + 45\cdot 97^{4} + 2\cdot 97^{5} + 72\cdot 97^{6} +O\left(97^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 30 + 94\cdot 97 + 65\cdot 97^{2} + 57\cdot 97^{3} + 66\cdot 97^{4} + 24\cdot 97^{5} + 34\cdot 97^{6} +O\left(97^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 31 + 18\cdot 97 + 83\cdot 97^{2} + 74\cdot 97^{3} + 25\cdot 97^{4} + 28\cdot 97^{5} + 96\cdot 97^{6} +O\left(97^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 45 + 25\cdot 97 + 60\cdot 97^{2} + 45\cdot 97^{3} + 65\cdot 97^{4} + 89\cdot 97^{5} + 90\cdot 97^{6} +O\left(97^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 52 + 71\cdot 97 + 36\cdot 97^{2} + 51\cdot 97^{3} + 31\cdot 97^{4} + 7\cdot 97^{5} + 6\cdot 97^{6} +O\left(97^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 66 + 78\cdot 97 + 13\cdot 97^{2} + 22\cdot 97^{3} + 71\cdot 97^{4} + 68\cdot 97^{5} +O\left(97^{ 7 }\right)$
$r_{ 7 }$	$=$	$ 67 + 2\cdot 97 + 31\cdot 97^{2} + 39\cdot 97^{3} + 30\cdot 97^{4} + 72\cdot 97^{5} + 62\cdot 97^{6} +O\left(97^{ 7 }\right)$
$r_{ 8 }$	$=$	$ 95 + 78\cdot 97 + 26\cdot 97^{2} + 39\cdot 97^{3} + 51\cdot 97^{4} + 94\cdot 97^{5} + 24\cdot 97^{6} +O\left(97^{ 7 }\right)$

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

Cycle notation

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

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

$(1,8)(4,5)$

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

$(2,7)(4,5)$

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.