Artin representation 15.113e10_8329e10.42t411.1c1

• Label: 15.113e10_8329e10.42t411.1c1
• Dimension: 15
• Group: $S_7$
• Conductor: $ 113^{10} \cdot 8329^{10}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$15$
Group:	$S_7$
Conductor:	$545397390487976174405872708445865484516716740728934230257649= 113^{10} \cdot 8329^{10} $
Artin number field:	Splitting field of $f= x^{7} - 2 x^{5} + x^{3} - x^{2} - x + 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	42T411
Parity:	Even
Determinant:	1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $ x^{2} + 42 x + 3 $

Roots:

$r_{ 1 }$	$=$	$ 16 a + 21 + \left(20 a + 10\right)\cdot 43 + \left(19 a + 21\right)\cdot 43^{2} + \left(42 a + 8\right)\cdot 43^{3} + \left(37 a + 15\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 25 + 30\cdot 43 + 19\cdot 43^{2} + 23\cdot 43^{3} + 25\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 27 a + 37 + \left(22 a + 14\right)\cdot 43 + \left(23 a + 20\right)\cdot 43^{2} + 31\cdot 43^{3} + \left(5 a + 10\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 16 + 40\cdot 43 + 21\cdot 43^{2} + 10\cdot 43^{3} + 28\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 15 + 28\cdot 43 + 24\cdot 43^{2} + 24\cdot 43^{3} + 13\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 13 a + 1 + \left(20 a + 20\right)\cdot 43 + \left(27 a + 28\right)\cdot 43^{2} + \left(19 a + 40\right)\cdot 43^{3} + \left(41 a + 6\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 30 a + 14 + \left(22 a + 27\right)\cdot 43 + \left(15 a + 35\right)\cdot 43^{2} + \left(23 a + 32\right)\cdot 43^{3} + \left(a + 28\right)\cdot 43^{4} +O\left(43^{ 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
$1$	$1$	$()$	$15$
$21$	$2$	$(1,2)$	$-5$
$105$	$2$	$(1,2)(3,4)(5,6)$	$3$
$105$	$2$	$(1,2)(3,4)$	$-1$
$70$	$3$	$(1,2,3)$	$3$
$280$	$3$	$(1,2,3)(4,5,6)$	$0$
$210$	$4$	$(1,2,3,4)$	$-1$
$630$	$4$	$(1,2,3,4)(5,6)$	$-1$
$504$	$5$	$(1,2,3,4,5)$	$0$
$210$	$6$	$(1,2,3)(4,5)(6,7)$	$-1$
$420$	$6$	$(1,2,3)(4,5)$	$1$
$840$	$6$	$(1,2,3,4,5,6)$	$0$
$720$	$7$	$(1,2,3,4,5,6,7)$	$1$
$504$	$10$	$(1,2,3,4,5)(6,7)$	$0$
$420$	$12$	$(1,2,3,4)(5,6,7)$	$-1$

The blue line marks the conjugacy class containing complex conjugation.