Artin representation 14.17e9_172519e9.30t565.1c1

• Label: 14.17e9_172519e9.30t565.1c1
• Dimension: 14
• Group: $S_7$
• Conductor: $ 17^{9} \cdot 172519^{9}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$14$
Group:	$S_7$
Conductor:	$16053609009208922465650590291068988511864659930724290992663= 17^{9} \cdot 172519^{9} $
Artin number field:	Splitting field of $f= x^{7} - x^{6} - 4 x^{3} + 2 x^{2} + 2 x - 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	30T565
Parity:	Odd
Determinant:	1.17_172519.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 181 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 181 }$: $ x^{2} + 177 x + 2 $

Roots:

$r_{ 1 }$	$=$	$ 7 + 156\cdot 181 + 139\cdot 181^{2} + 8\cdot 181^{3} + 138\cdot 181^{4} +O\left(181^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 96 a + 173 + \left(125 a + 54\right)\cdot 181 + \left(162 a + 138\right)\cdot 181^{2} + \left(78 a + 16\right)\cdot 181^{3} + \left(165 a + 175\right)\cdot 181^{4} +O\left(181^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 90 a + 173 + \left(95 a + 109\right)\cdot 181 + \left(121 a + 179\right)\cdot 181^{2} + \left(77 a + 36\right)\cdot 181^{3} + \left(29 a + 22\right)\cdot 181^{4} +O\left(181^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 87 + 168\cdot 181 + 177\cdot 181^{2} + 148\cdot 181^{3} + 59\cdot 181^{4} +O\left(181^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 100 + 95\cdot 181 + 121\cdot 181^{2} + 116\cdot 181^{3} + 51\cdot 181^{4} +O\left(181^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 85 a + 14 + \left(55 a + 99\right)\cdot 181 + \left(18 a + 120\right)\cdot 181^{2} + \left(102 a + 169\right)\cdot 181^{3} + \left(15 a + 33\right)\cdot 181^{4} +O\left(181^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 91 a + 171 + \left(85 a + 39\right)\cdot 181 + \left(59 a + 27\right)\cdot 181^{2} + \left(103 a + 45\right)\cdot 181^{3} + \left(151 a + 62\right)\cdot 181^{4} +O\left(181^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 7 }$

Cycle notation

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

$(1,2)$

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 7 }$	Character value
$1$	$1$	$()$	$14$
$21$	$2$	$(1,2)$	$-4$
$105$	$2$	$(1,2)(3,4)(5,6)$	$0$
$105$	$2$	$(1,2)(3,4)$	$2$
$70$	$3$	$(1,2,3)$	$-1$
$280$	$3$	$(1,2,3)(4,5,6)$	$2$
$210$	$4$	$(1,2,3,4)$	$2$
$630$	$4$	$(1,2,3,4)(5,6)$	$0$
$504$	$5$	$(1,2,3,4,5)$	$-1$
$210$	$6$	$(1,2,3)(4,5)(6,7)$	$-1$
$420$	$6$	$(1,2,3)(4,5)$	$-1$
$840$	$6$	$(1,2,3,4,5,6)$	$0$
$720$	$7$	$(1,2,3,4,5,6,7)$	$0$
$504$	$10$	$(1,2,3,4,5)(6,7)$	$1$
$420$	$12$	$(1,2,3,4)(5,6,7)$	$-1$

The blue line marks the conjugacy class containing complex conjugation.