Artin representation 6.23e5_10337e5.14t46.1c1

• Label: 6.23e5_10337e5.14t46.1c1
• Dimension: 6
• Group: $S_7$
• Conductor: $ 23^{5} \cdot 10337^{5}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$6$
Group:	$S_7$
Conductor:	$759646885244864506814313751= 23^{5} \cdot 10337^{5} $
Artin number field:	Splitting field of $f= x^{7} - x^{6} + x^{5} - x^{4} + x^{3} + x^{2} - 2 x + 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	14T46
Parity:	Odd
Determinant:	1.23_10337.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{2} + 38 x + 6 $

Roots:

$r_{ 1 }$	$=$	$ 30 a + 17 + \left(16 a + 20\right)\cdot 41 + \left(17 a + 22\right)\cdot 41^{2} + \left(14 a + 22\right)\cdot 41^{3} + \left(13 a + 8\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 34 a + 37 + \left(18 a + 38\right)\cdot 41 + \left(8 a + 25\right)\cdot 41^{2} + \left(17 a + 13\right)\cdot 41^{3} + \left(3 a + 29\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 35 + 31\cdot 41 + 28\cdot 41^{2} + 33\cdot 41^{3} + 6\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 7 a + 16 + \left(22 a + 20\right)\cdot 41 + \left(32 a + 32\right)\cdot 41^{2} + \left(23 a + 15\right)\cdot 41^{3} + \left(37 a + 22\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 22 + 36\cdot 41 + 29\cdot 41^{2} + 14\cdot 41^{3} + 24\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 13 + 16\cdot 41 + 7\cdot 41^{2} + 15\cdot 41^{3} + 38\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 11 a + 25 + \left(24 a + 40\right)\cdot 41 + \left(23 a + 16\right)\cdot 41^{2} + \left(26 a + 7\right)\cdot 41^{3} + \left(27 a + 34\right)\cdot 41^{4} +O\left(41^{ 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.