Artin representation 6.1567e5_2153e5.14t46.1c1

• Label: 6.1567e5_2153e5.14t46.1c1
• Dimension: 6
• Group: $S_7$
• Conductor: $ 1567^{5} \cdot 2153^{5}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$6$
Group:	$S_7$
Conductor:	$437084223995624533462923469993751= 1567^{5} \cdot 2153^{5} $
Artin number field:	Splitting field of $f= x^{7} - 2 x^{5} - 2 x^{4} - x^{3} + 3 x^{2} + x - 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	14T46
Parity:	Odd
Determinant:	1.1567_2153.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$: $ x^{2} + 69 x + 7 $

Roots:

$r_{ 1 }$	$=$	$ 31 + 11\cdot 71 + 57\cdot 71^{2} + 70\cdot 71^{3} + 27\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 67 a + 31 + \left(13 a + 27\right)\cdot 71 + \left(61 a + 3\right)\cdot 71^{2} + \left(15 a + 47\right)\cdot 71^{3} + \left(27 a + 15\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 23 + 52\cdot 71 + 28\cdot 71^{2} + 57\cdot 71^{3} + 2\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 41 + 36\cdot 71 + 2\cdot 71^{2} + 62\cdot 71^{3} + 49\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 4 a + 23 + \left(57 a + 59\right)\cdot 71 + \left(9 a + 40\right)\cdot 71^{2} + \left(55 a + 17\right)\cdot 71^{3} + \left(43 a + 54\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 64 a + 39 + \left(37 a + 42\right)\cdot 71 + \left(48 a + 10\right)\cdot 71^{2} + \left(17 a + 21\right)\cdot 71^{3} + \left(67 a + 8\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 7 a + 25 + \left(33 a + 54\right)\cdot 71 + \left(22 a + 69\right)\cdot 71^{2} + \left(53 a + 7\right)\cdot 71^{3} + \left(3 a + 54\right)\cdot 71^{4} +O\left(71^{ 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.