Artin representation 4.2e17_5e2.8t21.2c1

• Label: 4.2e17_5e2.8t21.2c1
• Dimension: 4
• Group: $C_2^3 : C_4 $
• Conductor: $ 2^{17} \cdot 5^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : C_4 $
Conductor:	$3276800= 2^{17} \cdot 5^{2} $
Artin number field:	Splitting field of $f= x^{8} - 8 x^{6} + 12 x^{4} + 20 x^{2} + 25 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$C_2^3: C_4$
Parity:	Even
Determinant:	1.2e3.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 313 }$ to precision 7.

Roots:

$r_{ 1 }$	$=$	$ 63 + 87\cdot 313 + 109\cdot 313^{2} + 209\cdot 313^{3} + 153\cdot 313^{4} + 74\cdot 313^{5} + 145\cdot 313^{6} +O\left(313^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 122 + 134\cdot 313 + 60\cdot 313^{2} + 174\cdot 313^{3} + 36\cdot 313^{4} + 23\cdot 313^{5} + 186\cdot 313^{6} +O\left(313^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 140 + 120\cdot 313 + 34\cdot 313^{2} + 12\cdot 313^{3} + 2\cdot 313^{4} + 128\cdot 313^{5} + 143\cdot 313^{6} +O\left(313^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 145 + 74\cdot 313 + 97\cdot 313^{2} + 62\cdot 313^{3} + 243\cdot 313^{4} + 83\cdot 313^{5} + 99\cdot 313^{6} +O\left(313^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 168 + 238\cdot 313 + 215\cdot 313^{2} + 250\cdot 313^{3} + 69\cdot 313^{4} + 229\cdot 313^{5} + 213\cdot 313^{6} +O\left(313^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 173 + 192\cdot 313 + 278\cdot 313^{2} + 300\cdot 313^{3} + 310\cdot 313^{4} + 184\cdot 313^{5} + 169\cdot 313^{6} +O\left(313^{ 7 }\right)$
$r_{ 7 }$	$=$	$ 191 + 178\cdot 313 + 252\cdot 313^{2} + 138\cdot 313^{3} + 276\cdot 313^{4} + 289\cdot 313^{5} + 126\cdot 313^{6} +O\left(313^{ 7 }\right)$
$r_{ 8 }$	$=$	$ 250 + 225\cdot 313 + 203\cdot 313^{2} + 103\cdot 313^{3} + 159\cdot 313^{4} + 238\cdot 313^{5} + 167\cdot 313^{6} +O\left(313^{ 7 }\right)$

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

Cycle notation

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

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.