Artin representation 6.463_9689.7t7.1c1

• Label: 6.463_9689.7t7.1c1
• Dimension: 6
• Group: $S_7$
• Conductor: $ 463 \cdot 9689 $
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$6$
Group:	$S_7$
Conductor:	$4486007= 463 \cdot 9689 $
Artin number field:	Splitting field of $f= x^{7} - 2 x^{6} - x^{5} + 6 x^{4} - 2 x^{3} - 5 x^{2} + x + 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$S_7$
Parity:	Odd
Determinant:	1.463_9689.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 139 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 139 }$: $ x^{2} + 138 x + 2 $

Roots:

$r_{ 1 }$	$=$	$ 59 a + 67 + \left(79 a + 78\right)\cdot 139 + \left(117 a + 23\right)\cdot 139^{2} + \left(53 a + 126\right)\cdot 139^{3} + \left(15 a + 101\right)\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 129 a + 91 + \left(96 a + 120\right)\cdot 139 + \left(87 a + 17\right)\cdot 139^{2} + \left(73 a + 5\right)\cdot 139^{3} + \left(78 a + 49\right)\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 79 + 51\cdot 139 + 101\cdot 139^{2} + 75\cdot 139^{3} + 137\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 80 a + 126 + \left(59 a + 98\right)\cdot 139 + \left(21 a + 61\right)\cdot 139^{2} + \left(85 a + 62\right)\cdot 139^{3} + \left(123 a + 63\right)\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 74 + 18\cdot 139 + 71\cdot 139^{2} + 56\cdot 139^{3} + 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 40 + 99\cdot 139 + 132\cdot 139^{2} + 99\cdot 139^{3} + 9\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 10 a + 81 + \left(42 a + 88\right)\cdot 139 + \left(51 a + 8\right)\cdot 139^{2} + \left(65 a + 130\right)\cdot 139^{3} + \left(60 a + 53\right)\cdot 139^{4} +O\left(139^{ 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
$1$	$1$	$()$	$6$
$21$	$2$	$(1,2)$	$4$
$105$	$2$	$(1,2)(3,4)(5,6)$	$0$
$105$	$2$	$(1,2)(3,4)$	$2$
$70$	$3$	$(1,2,3)$	$3$
$280$	$3$	$(1,2,3)(4,5,6)$	$0$
$210$	$4$	$(1,2,3,4)$	$2$
$630$	$4$	$(1,2,3,4)(5,6)$	$0$
$504$	$5$	$(1,2,3,4,5)$	$1$
$210$	$6$	$(1,2,3)(4,5)(6,7)$	$-1$
$420$	$6$	$(1,2,3)(4,5)$	$1$
$840$	$6$	$(1,2,3,4,5,6)$	$0$
$720$	$7$	$(1,2,3,4,5,6,7)$	$-1$
$504$	$10$	$(1,2,3,4,5)(6,7)$	$-1$
$420$	$12$	$(1,2,3,4)(5,6,7)$	$-1$

The blue line marks the conjugacy class containing complex conjugation.