Artin representation 2.2e7_7.8t17.3c2

• Label: 2.2e7_7.8t17.3c2
• Dimension: 2
• Group: $C_4\wr C_2$
• Conductor: $ 2^{7} \cdot 7 $
• Root number: not computed
• Frobenius-Schur indicator: 0

Basic invariants

Dimension:	$2$
Group:	$C_4\wr C_2$
Conductor:	$896= 2^{7} \cdot 7 $
Artin number field:	Splitting field of $f= x^{8} - 2 x^{6} + 6 x^{4} - 12 x^{2} + 8 $ over $\Q$
Size of Galois orbit:	2
Smallest containing permutation representation:	$C_4\wr C_2$
Parity:	Odd
Determinant:	1.2e4_7.4t1.2c2

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 239 }$ to precision 7.

Roots:

$r_{ 1 }$	$=$	$ 23 + 157\cdot 239 + 183\cdot 239^{2} + 64\cdot 239^{3} + 132\cdot 239^{4} + 95\cdot 239^{5} + 129\cdot 239^{6} +O\left(239^{ 7 }\right)$
$r_{ 2 }$	$=$	$ 45 + 235\cdot 239 + 226\cdot 239^{2} + 182\cdot 239^{3} + 232\cdot 239^{4} + 9\cdot 239^{5} + 31\cdot 239^{6} +O\left(239^{ 7 }\right)$
$r_{ 3 }$	$=$	$ 64 + 233\cdot 239 + 227\cdot 239^{2} + 16\cdot 239^{3} + 81\cdot 239^{4} + 184\cdot 239^{5} + 35\cdot 239^{6} +O\left(239^{ 7 }\right)$
$r_{ 4 }$	$=$	$ 98 + 210\cdot 239 + 114\cdot 239^{2} + 82\cdot 239^{3} + 36\cdot 239^{4} + 55\cdot 239^{5} + 199\cdot 239^{6} +O\left(239^{ 7 }\right)$
$r_{ 5 }$	$=$	$ 141 + 28\cdot 239 + 124\cdot 239^{2} + 156\cdot 239^{3} + 202\cdot 239^{4} + 183\cdot 239^{5} + 39\cdot 239^{6} +O\left(239^{ 7 }\right)$
$r_{ 6 }$	$=$	$ 175 + 5\cdot 239 + 11\cdot 239^{2} + 222\cdot 239^{3} + 157\cdot 239^{4} + 54\cdot 239^{5} + 203\cdot 239^{6} +O\left(239^{ 7 }\right)$
$r_{ 7 }$	$=$	$ 194 + 3\cdot 239 + 12\cdot 239^{2} + 56\cdot 239^{3} + 6\cdot 239^{4} + 229\cdot 239^{5} + 207\cdot 239^{6} +O\left(239^{ 7 }\right)$
$r_{ 8 }$	$=$	$ 216 + 81\cdot 239 + 55\cdot 239^{2} + 174\cdot 239^{3} + 106\cdot 239^{4} + 143\cdot 239^{5} + 109\cdot 239^{6} +O\left(239^{ 7 }\right)$

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

Cycle notation

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

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

$(1,5,8,4)$

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.