Artin representation 4.5e4_11e2_41e2.8t16.4c1

• Label: 4.5e4_11e2_41e2.8t16.4c1
• Dimension: 4
• Group: $(C_8:C_2):C_2$
• Conductor: $ 5^{4} \cdot 11^{2} \cdot 41^{2}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$(C_8:C_2):C_2$
Conductor:	$127125625= 5^{4} \cdot 11^{2} \cdot 41^{2} $
Artin number field:	Splitting field of $f= x^{8} - 2 x^{7} - 32 x^{6} + 41 x^{5} + 380 x^{4} - 249 x^{3} - 1892 x^{2} + 483 x + 3371 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$(C_8:C_2):C_2$
Parity:	Even
Determinant:	1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 211 }$ to precision 8.

Roots:

$r_{ 1 }$	$=$	$ 10 + 142\cdot 211 + 77\cdot 211^{2} + 65\cdot 211^{3} + 51\cdot 211^{4} + 109\cdot 211^{5} + 137\cdot 211^{6} + 208\cdot 211^{7} +O\left(211^{ 8 }\right)$
$r_{ 2 }$	$=$	$ 11 + 185\cdot 211 + 75\cdot 211^{2} + 176\cdot 211^{3} + 124\cdot 211^{4} + 93\cdot 211^{5} + 178\cdot 211^{6} + 67\cdot 211^{7} +O\left(211^{ 8 }\right)$
$r_{ 3 }$	$=$	$ 22 + 172\cdot 211 + 118\cdot 211^{2} + 18\cdot 211^{3} + 37\cdot 211^{4} + 167\cdot 211^{5} + 61\cdot 211^{6} + 153\cdot 211^{7} +O\left(211^{ 8 }\right)$
$r_{ 4 }$	$=$	$ 52 + 200\cdot 211 + 155\cdot 211^{2} + 90\cdot 211^{3} + 54\cdot 211^{4} + 145\cdot 211^{5} + 145\cdot 211^{6} + 71\cdot 211^{7} +O\left(211^{ 8 }\right)$
$r_{ 5 }$	$=$	$ 62 + 50\cdot 211 + 191\cdot 211^{2} + 55\cdot 211^{3} + 29\cdot 211^{4} + 175\cdot 211^{5} + 34\cdot 211^{6} + 182\cdot 211^{7} +O\left(211^{ 8 }\right)$
$r_{ 6 }$	$=$	$ 117 + 14\cdot 211 + 36\cdot 211^{2} + 171\cdot 211^{3} + 19\cdot 211^{4} + 197\cdot 211^{5} + 146\cdot 211^{6} + 18\cdot 211^{7} +O\left(211^{ 8 }\right)$
$r_{ 7 }$	$=$	$ 169 + 133\cdot 211 + 149\cdot 211^{2} + 161\cdot 211^{3} + 208\cdot 211^{4} + 51\cdot 211^{5} + 44\cdot 211^{6} + 203\cdot 211^{7} +O\left(211^{ 8 }\right)$
$r_{ 8 }$	$=$	$ 192 + 156\cdot 211 + 38\cdot 211^{2} + 104\cdot 211^{3} + 107\cdot 211^{4} + 115\cdot 211^{5} + 94\cdot 211^{6} + 149\cdot 211^{7} +O\left(211^{ 8 }\right)$

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

Cycle notation

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

$(2,3)(5,6)$

$(4,8)(5,6)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.