Artin representation 4.2e6_3e2_5e2_7e2.8t22.18c1

• Label: 4.2e6_3e2_5e2_7e2.8t22.18c1
• Dimension: 4
• Group: $C_2^3 : D_4 $
• Conductor: $ 2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : D_4 $
Conductor:	$705600= 2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} $
Artin number field:	Splitting field of $f= x^{8} - 3 x^{7} - 2 x^{6} + 8 x^{5} + 11 x^{4} - 20 x^{3} - 12 x^{2} + 7 x + 19 $ 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 $\Q_{ 331 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 14 + 30\cdot 331 + 254\cdot 331^{2} + 72\cdot 331^{3} + 120\cdot 331^{4} +O\left(331^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 90 + 149\cdot 331 + 331^{2} + 284\cdot 331^{3} + 314\cdot 331^{4} +O\left(331^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 135 + 59\cdot 331 + 41\cdot 331^{2} + 90\cdot 331^{3} + 318\cdot 331^{4} +O\left(331^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 239 + 300\cdot 331 + 268\cdot 331^{2} + 224\cdot 331^{3} + 223\cdot 331^{4} +O\left(331^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 268 + 139\cdot 331 + 43\cdot 331^{2} + 82\cdot 331^{3} + 294\cdot 331^{4} +O\left(331^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 287 + 256\cdot 331 + 118\cdot 331^{2} + 165\cdot 331^{3} + 196\cdot 331^{4} +O\left(331^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 304 + 288\cdot 331 + 145\cdot 331^{2} + 108\cdot 331^{3} + 251\cdot 331^{4} +O\left(331^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 321 + 98\cdot 331 + 119\cdot 331^{2} + 296\cdot 331^{3} + 266\cdot 331^{4} +O\left(331^{ 5 }\right)$

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

Cycle notation

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

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

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

$(1,7)(4,8)$

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.