Artin representation 6.149e5_5981e5.14t46.1c1

• Label: 6.149e5_5981e5.14t46.1c1
• Dimension: 6
• Group: $S_7$
• Conductor: $ 149^{5} \cdot 5981^{5}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$6$
Group:	$S_7$
Conductor:	$562082875380053796085510546849= 149^{5} \cdot 5981^{5} $
Artin number field:	Splitting field of $f= x^{7} - x^{6} - x^{5} + 2 x^{4} - 3 x^{2} + 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	14T46
Parity:	Even
Determinant:	1.149_5981.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 137 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 137 }$: $ x^{2} + 131 x + 3 $

Roots:

$r_{ 1 }$	$=$	$ 68 a + 98 + \left(51 a + 90\right)\cdot 137 + \left(a + 72\right)\cdot 137^{2} + \left(107 a + 45\right)\cdot 137^{3} + \left(57 a + 83\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 107 + 12\cdot 137 + 27\cdot 137^{2} + 70\cdot 137^{3} + 101\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 120 a + 122 + \left(78 a + 49\right)\cdot 137 + \left(73 a + 112\right)\cdot 137^{2} + \left(115 a + 34\right)\cdot 137^{3} + \left(64 a + 89\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 49 + 44\cdot 137 + 17\cdot 137^{2} + 110\cdot 137^{3} + 47\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 69 a + 95 + \left(85 a + 57\right)\cdot 137 + \left(135 a + 29\right)\cdot 137^{2} + \left(29 a + 1\right)\cdot 137^{3} + \left(79 a + 49\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 58 + 26\cdot 137 + 88\cdot 137^{2} + 42\cdot 137^{3} + 88\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 17 a + 20 + \left(58 a + 129\right)\cdot 137 + \left(63 a + 63\right)\cdot 137^{2} + \left(21 a + 106\right)\cdot 137^{3} + \left(72 a + 88\right)\cdot 137^{4} +O\left(137^{ 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.