Artin representation 4.2e10_3e2_5e2.8t22.3c1

• Label: 4.2e10_3e2_5e2.8t22.3c1
• Dimension: 4
• Group: $C_2^3 : D_4 $
• Conductor: $ 2^{10} \cdot 3^{2} \cdot 5^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : D_4 $
Conductor:	$230400= 2^{10} \cdot 3^{2} \cdot 5^{2} $
Artin number field:	Splitting field of $f= x^{8} - 4 x^{6} - 8 x^{5} + x^{4} + 16 x^{3} + 22 x^{2} + 12 x + 3 $ 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 $\Q_{ 241 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 39 + 35\cdot 241 + 59\cdot 241^{2} + 2\cdot 241^{3} + 189\cdot 241^{4} +O\left(241^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 51 + 38\cdot 241 + 152\cdot 241^{2} + 189\cdot 241^{3} + 237\cdot 241^{4} +O\left(241^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 76 + 23\cdot 241 + 115\cdot 241^{2} + 80\cdot 241^{3} + 166\cdot 241^{4} +O\left(241^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 102 + 77\cdot 241 + 90\cdot 241^{2} + 3\cdot 241^{3} + 53\cdot 241^{4} +O\left(241^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 144 + 20\cdot 241 + 2\cdot 241^{2} + 160\cdot 241^{3} + 33\cdot 241^{4} +O\left(241^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 145 + 170\cdot 241 + 158\cdot 241^{2} + 56\cdot 241^{3} + 162\cdot 241^{4} +O\left(241^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 185 + 104\cdot 241 + 237\cdot 241^{2} + 128\cdot 241^{3} + 157\cdot 241^{4} +O\left(241^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 222 + 11\cdot 241 + 149\cdot 241^{2} + 101\cdot 241^{3} + 205\cdot 241^{4} +O\left(241^{ 5 }\right)$

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

Cycle notation

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

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

$(1,3)(2,4)$

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

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

$(2,4)(6,8)$

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.