Artin representation 4.2e12_3e2_5e2.8t22.10c1

• Label: 4.2e12_3e2_5e2.8t22.10c1
• Dimension: 4
• Group: $C_2^3 : D_4 $
• Conductor: $ 2^{12} \cdot 3^{2} \cdot 5^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : D_4 $
Conductor:	$921600= 2^{12} \cdot 3^{2} \cdot 5^{2} $
Artin number field:	Splitting field of $f= x^{8} - 4 x^{6} + 13 x^{4} - 18 x^{2} + 9 $ 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 $\Q_{ 409 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 30 + 353\cdot 409 + 252\cdot 409^{2} + 199\cdot 409^{3} + 383\cdot 409^{4} +O\left(409^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 90 + 14\cdot 409 + 238\cdot 409^{2} + 285\cdot 409^{3} + 333\cdot 409^{4} +O\left(409^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 133 + 26\cdot 409 + 206\cdot 409^{2} + 273\cdot 409^{3} + 364\cdot 409^{4} +O\left(409^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 203 + 172\cdot 409 + 330\cdot 409^{2} + 219\cdot 409^{3} + 186\cdot 409^{4} +O\left(409^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 206 + 236\cdot 409 + 78\cdot 409^{2} + 189\cdot 409^{3} + 222\cdot 409^{4} +O\left(409^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 276 + 382\cdot 409 + 202\cdot 409^{2} + 135\cdot 409^{3} + 44\cdot 409^{4} +O\left(409^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 319 + 394\cdot 409 + 170\cdot 409^{2} + 123\cdot 409^{3} + 75\cdot 409^{4} +O\left(409^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 379 + 55\cdot 409 + 156\cdot 409^{2} + 209\cdot 409^{3} + 25\cdot 409^{4} +O\left(409^{ 5 }\right)$

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

Cycle notation

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

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

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

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.