Galois orbit of Artin representations 2.5e2_11e2.8t17.1

• Label: 2.5e2_11e2.8t17.1
• Dimension: 2
• Group: $C_4\wr C_2$
• Conductor: $ 5^{2} \cdot 11^{2}$
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$2$
Group:	$C_4\wr C_2$
Conductor:	$3025= 5^{2} \cdot 11^{2} $
Artin number field:	Splitting field of $f= x^{8} - 3 x^{7} + 7 x^{6} - 15 x^{5} + 18 x^{4} - 16 x^{3} + 13 x^{2} - 5 x + 3 $ 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 $\Q_{ 71 }$ to precision 6.

Roots:

$r_{ 1 }$	$=$	$ 15 + 45\cdot 71 + 20\cdot 71^{2} + 41\cdot 71^{3} + 39\cdot 71^{4} + 39\cdot 71^{5} +O\left(71^{ 6 }\right)$
$r_{ 2 }$	$=$	$ 16 + 41\cdot 71 + 48\cdot 71^{2} + 50\cdot 71^{3} + 2\cdot 71^{4} + 46\cdot 71^{5} +O\left(71^{ 6 }\right)$
$r_{ 3 }$	$=$	$ 27 + 34\cdot 71 + 44\cdot 71^{2} + 59\cdot 71^{3} + 44\cdot 71^{4} + 21\cdot 71^{5} +O\left(71^{ 6 }\right)$
$r_{ 4 }$	$=$	$ 33 + 11\cdot 71 + 40\cdot 71^{2} + 12\cdot 71^{3} + 31\cdot 71^{4} + 41\cdot 71^{5} +O\left(71^{ 6 }\right)$
$r_{ 5 }$	$=$	$ 35 + 42\cdot 71 + 25\cdot 71^{2} + 15\cdot 71^{3} + 18\cdot 71^{4} + 9\cdot 71^{5} +O\left(71^{ 6 }\right)$
$r_{ 6 }$	$=$	$ 36 + 16\cdot 71 + 5\cdot 71^{2} + 53\cdot 71^{3} + 39\cdot 71^{4} + 31\cdot 71^{5} +O\left(71^{ 6 }\right)$
$r_{ 7 }$	$=$	$ 62 + 43\cdot 71 + 58\cdot 71^{2} + 42\cdot 71^{3} + 29\cdot 71^{4} + 54\cdot 71^{5} +O\left(71^{ 6 }\right)$
$r_{ 8 }$	$=$	$ 63 + 48\cdot 71 + 40\cdot 71^{2} + 8\cdot 71^{3} + 7\cdot 71^{4} + 40\cdot 71^{5} +O\left(71^{ 6 }\right)$

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

Cycle notation

$(3,8)(4,5)$

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

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

$(3,4,8,5)$

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.