Artin representation 4.5e3_19e4.8t30.3c1

• Label: 4.5e3_19e4.8t30.3c1
• Dimension: 4
• Group: $(((C_4 \times C_2): C_2):C_2):C_2$
• Conductor: $ 5^{3} \cdot 19^{4}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$(((C_4 \times C_2): C_2):C_2):C_2$
Conductor:	$16290125= 5^{3} \cdot 19^{4} $
Artin number field:	Splitting field of $f= x^{8} - x^{7} + 6 x^{5} - 11 x^{4} + 6 x^{3} - x + 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$(((C_4 \times C_2): C_2):C_2):C_2$
Parity:	Even
Determinant:	1.5.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 191 }$ to precision 8.

Roots:

$r_{ 1 }$	$=$	$ 2 + 52\cdot 191 + 59\cdot 191^{2} + 187\cdot 191^{3} + 107\cdot 191^{4} + 74\cdot 191^{5} + 114\cdot 191^{6} + 40\cdot 191^{7} +O\left(191^{ 8 }\right)$
$r_{ 2 }$	$=$	$ 10 + 158\cdot 191 + 6\cdot 191^{2} + 20\cdot 191^{3} + 91\cdot 191^{4} + 32\cdot 191^{5} + 15\cdot 191^{6} + 131\cdot 191^{7} +O\left(191^{ 8 }\right)$
$r_{ 3 }$	$=$	$ 76 + 7\cdot 191 + 108\cdot 191^{2} + 113\cdot 191^{3} + 82\cdot 191^{4} + 121\cdot 191^{5} + 142\cdot 191^{6} + 12\cdot 191^{7} +O\left(191^{ 8 }\right)$
$r_{ 4 }$	$=$	$ 93 + 191 + 21\cdot 191^{2} + 36\cdot 191^{3} + 40\cdot 191^{4} + 171\cdot 191^{5} + 106\cdot 191^{6} + 137\cdot 191^{7} +O\left(191^{ 8 }\right)$
$r_{ 5 }$	$=$	$ 96 + 82\cdot 191 + 84\cdot 191^{2} + 51\cdot 191^{3} + 34\cdot 191^{4} + 67\cdot 191^{5} + 114\cdot 191^{6} + 65\cdot 191^{7} +O\left(191^{ 8 }\right)$
$r_{ 6 }$	$=$	$ 140 + 162\cdot 191 + 125\cdot 191^{2} + 24\cdot 191^{3} + 164\cdot 191^{4} + 141\cdot 191^{5} + 185\cdot 191^{6} + 107\cdot 191^{7} +O\left(191^{ 8 }\right)$
$r_{ 7 }$	$=$	$ 172 + 51\cdot 191 + 127\cdot 191^{2} + 39\cdot 191^{3} + 168\cdot 191^{4} + 166\cdot 191^{5} + 12\cdot 191^{6} + 138\cdot 191^{7} +O\left(191^{ 8 }\right)$
$r_{ 8 }$	$=$	$ 176 + 56\cdot 191 + 40\cdot 191^{2} + 100\cdot 191^{3} + 75\cdot 191^{4} + 179\cdot 191^{5} + 71\cdot 191^{6} + 130\cdot 191^{7} +O\left(191^{ 8 }\right)$

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

Cycle notation

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

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

$(2,7)(6,8)$

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

$(3,4)(6,8)$

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.