Galois orbit of Artin representations 2.2e2_5e2_13e2.8t17.1

• Label: 2.2e2_5e2_13e2.8t17.1
• Dimension: 2
• Group: $C_4\wr C_2$
• Conductor: $ 2^{2} \cdot 5^{2} \cdot 13^{2}$
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$2$
Group:	$C_4\wr C_2$
Conductor:	$16900= 2^{2} \cdot 5^{2} \cdot 13^{2} $
Artin number field:	Splitting field of $f= x^{8} - 2 x^{7} + 4 x^{6} - 8 x^{5} + 10 x^{4} - 12 x^{3} + 19 x^{2} - 8 x + 1 $ over $\Q$
Size of Galois orbit:	2
Smallest containing permutation representation:	$C_4\wr C_2$
Parity:	Odd

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 137 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 8 + 78\cdot 137 + 12\cdot 137^{2} + 111\cdot 137^{3} + 82\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 25 + 26\cdot 137 + 115\cdot 137^{2} + 123\cdot 137^{3} + 8\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 53 + 54\cdot 137 + 66\cdot 137^{2} + 107\cdot 137^{3} + 98\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 56 + 46\cdot 137 + 93\cdot 137^{2} + 60\cdot 137^{3} + 30\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 83 + 100\cdot 137 + 24\cdot 137^{2} + 2\cdot 137^{3} + 87\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 87 + 88\cdot 137 + 11\cdot 137^{2} + 4\cdot 137^{3} + 135\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 104 + 98\cdot 137 + 48\cdot 137^{2} + 54\cdot 137^{3} + 105\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 134 + 54\cdot 137 + 38\cdot 137^{2} + 84\cdot 137^{3} + 136\cdot 137^{4} +O\left(137^{ 5 }\right)$

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

Cycle notation

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

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

$(1,8)(5,6)$

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

$(1,5,8,6)$

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 8 }$	Character values
			$c1$	$c2$
$1$	$1$	$()$	$2$	$2$
$1$	$2$	$(1,8)(2,4)(3,7)(5,6)$	$-2$	$-2$
$2$	$2$	$(1,8)(5,6)$	$0$	$0$
$4$	$2$	$(1,4)(2,8)(3,5)(6,7)$	$0$	$0$
$1$	$4$	$(1,5,8,6)(2,7,4,3)$	$-2 \zeta_{4}$	$2 \zeta_{4}$
$1$	$4$	$(1,6,8,5)(2,3,4,7)$	$2 \zeta_{4}$	$-2 \zeta_{4}$
$2$	$4$	$(1,5,8,6)(2,3,4,7)$	$0$	$0$
$2$	$4$	$(1,5,8,6)$	$\zeta_{4} - 1$	$-\zeta_{4} - 1$
$2$	$4$	$(1,6,8,5)$	$-\zeta_{4} - 1$	$\zeta_{4} - 1$
$2$	$4$	$(1,8)(2,3,4,7)(5,6)$	$-\zeta_{4} + 1$	$\zeta_{4} + 1$
$2$	$4$	$(1,8)(2,7,4,3)(5,6)$	$\zeta_{4} + 1$	$-\zeta_{4} + 1$
$4$	$4$	$(1,4,8,2)(3,6,7,5)$	$0$	$0$
$4$	$8$	$(1,4,5,3,8,2,6,7)$	$0$	$0$
$4$	$8$	$(1,3,6,4,8,7,5,2)$	$0$	$0$

The blue line marks the conjugacy class containing complex conjugation.