Artin representation 4.5e3_7e2_11e2.8t21.4c1

• Label: 4.5e3_7e2_11e2.8t21.4c1
• Dimension: 4
• Group: $C_2^3 : C_4 $
• Conductor: $ 5^{3} \cdot 7^{2} \cdot 11^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : C_4 $
Conductor:	$741125= 5^{3} \cdot 7^{2} \cdot 11^{2} $
Artin number field:	Splitting field of $f= x^{8} - 4 x^{7} + 14 x^{5} - x^{4} - 26 x^{3} - 5 x^{2} + 21 x + 26 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$C_2^3: C_4$
Parity:	Even
Determinant:	1.5.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 89 }$ to precision 9.

Roots:

$r_{ 1 }$	$=$	$ 23 + 85\cdot 89 + 57\cdot 89^{2} + 33\cdot 89^{3} + 21\cdot 89^{4} + 43\cdot 89^{5} + 55\cdot 89^{6} + 7\cdot 89^{7} + 84\cdot 89^{8} +O\left(89^{ 9 }\right)$
$r_{ 2 }$	$=$	$ 30 + 33\cdot 89 + 69\cdot 89^{2} + 27\cdot 89^{3} + 62\cdot 89^{5} + 73\cdot 89^{6} + 37\cdot 89^{7} + 31\cdot 89^{8} +O\left(89^{ 9 }\right)$
$r_{ 3 }$	$=$	$ 42 + 78\cdot 89 + 84\cdot 89^{2} + 40\cdot 89^{3} + 89^{4} + 20\cdot 89^{5} + 19\cdot 89^{6} + 70\cdot 89^{7} + 4\cdot 89^{8} +O\left(89^{ 9 }\right)$
$r_{ 4 }$	$=$	$ 44 + 10\cdot 89 + 65\cdot 89^{2} + 47\cdot 89^{3} + 89^{4} + 60\cdot 89^{5} + 76\cdot 89^{6} + 46\cdot 89^{7} + 54\cdot 89^{8} +O\left(89^{ 9 }\right)$
$r_{ 5 }$	$=$	$ 46 + 78\cdot 89 + 23\cdot 89^{2} + 41\cdot 89^{3} + 87\cdot 89^{4} + 28\cdot 89^{5} + 12\cdot 89^{6} + 42\cdot 89^{7} + 34\cdot 89^{8} +O\left(89^{ 9 }\right)$
$r_{ 6 }$	$=$	$ 48 + 10\cdot 89 + 4\cdot 89^{2} + 48\cdot 89^{3} + 87\cdot 89^{4} + 68\cdot 89^{5} + 69\cdot 89^{6} + 18\cdot 89^{7} + 84\cdot 89^{8} +O\left(89^{ 9 }\right)$
$r_{ 7 }$	$=$	$ 60 + 55\cdot 89 + 19\cdot 89^{2} + 61\cdot 89^{3} + 88\cdot 89^{4} + 26\cdot 89^{5} + 15\cdot 89^{6} + 51\cdot 89^{7} + 57\cdot 89^{8} +O\left(89^{ 9 }\right)$
$r_{ 8 }$	$=$	$ 67 + 3\cdot 89 + 31\cdot 89^{2} + 55\cdot 89^{3} + 67\cdot 89^{4} + 45\cdot 89^{5} + 33\cdot 89^{6} + 81\cdot 89^{7} + 4\cdot 89^{8} +O\left(89^{ 9 }\right)$

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

Cycle notation

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

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

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

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

$(1,8)(3,6)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.