Artin representation 4.2e4_3e2_5e2_7e2.8t22.3c1

• Label: 4.2e4_3e2_5e2_7e2.8t22.3c1
• Dimension: 4
• Group: $C_2^3 : D_4 $
• Conductor: $ 2^{4} \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:	$176400= 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} $
Artin number field:	Splitting field of $f= x^{8} - 2 x^{7} + 7 x^{6} - 2 x^{5} - 4 x^{3} + 3 x^{2} + 4 x + 1 $ 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 $\Q_{ 421 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 3 + 393\cdot 421 + 221\cdot 421^{2} + 238\cdot 421^{3} + 273\cdot 421^{4} +O\left(421^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 24 + 51\cdot 421 + 172\cdot 421^{2} + 285\cdot 421^{3} + 335\cdot 421^{4} +O\left(421^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 44 + 265\cdot 421 + 165\cdot 421^{2} + 250\cdot 421^{3} + 371\cdot 421^{4} +O\left(421^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 124 + 404\cdot 421 + 16\cdot 421^{2} + 228\cdot 421^{3} + 140\cdot 421^{4} +O\left(421^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 162 + 248\cdot 421 + 110\cdot 421^{2} + 408\cdot 421^{3} + 42\cdot 421^{4} +O\left(421^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 227 + 318\cdot 421 + 208\cdot 421^{2} + 207\cdot 421^{3} + 250\cdot 421^{4} +O\left(421^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 271 + 414\cdot 421 + 9\cdot 421^{2} + 90\cdot 421^{3} + 92\cdot 421^{4} +O\left(421^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 410 + 9\cdot 421 + 357\cdot 421^{2} + 396\cdot 421^{3} + 176\cdot 421^{4} +O\left(421^{ 5 }\right)$

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

Cycle notation

$(2,4)(3,5)$

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

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

$(1,7)(2,4)$

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.