Artin representation 35.163e15_6883e15.70.1c1

• Label: 35.163e15_6883e15.70.1c1
• Dimension: 35
• Group: $S_7$
• Conductor: $ 163^{15} \cdot 6883^{15}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$35$
Group:	$S_7$
Conductor:	$5616692000600651288034575315018913012058552593099707439628474114201877582177002713579899449= 163^{15} \cdot 6883^{15} $
Artin number field:	Splitting field of $f= x^{7} - x^{6} - x^{5} - x^{4} + 2 x^{3} - 2 x^{2} + 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	70
Parity:	Even
Determinant:	1.163_6883.2t1.1c1

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:

$r_{ 1 }$	$=$	$ 34 a + 14 + \left(37 a + 33\right)\cdot 41 + \left(23 a + 36\right)\cdot 41^{2} + \left(13 a + 13\right)\cdot 41^{3} + \left(26 a + 5\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 27 + 22\cdot 41 + 31\cdot 41^{2} + 5\cdot 41^{3} + 17\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 7 a + 34 + \left(3 a + 30\right)\cdot 41 + \left(17 a + 29\right)\cdot 41^{2} + \left(27 a + 30\right)\cdot 41^{3} + \left(14 a + 29\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 19 + 41 + 38\cdot 41^{2} + 22\cdot 41^{3} + 20\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 31 a + 3 + \left(6 a + 8\right)\cdot 41 + \left(32 a + 5\right)\cdot 41^{2} + \left(2 a + 17\right)\cdot 41^{3} + \left(14 a + 27\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 10 a + 14 + \left(34 a + 38\right)\cdot 41 + \left(8 a + 12\right)\cdot 41^{2} + \left(38 a + 34\right)\cdot 41^{3} + \left(26 a + 25\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 13 + 29\cdot 41 + 9\cdot 41^{2} + 39\cdot 41^{3} + 37\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
$1$	$1$	$()$	$35$
$21$	$2$	$(1,2)$	$5$
$105$	$2$	$(1,2)(3,4)(5,6)$	$1$
$105$	$2$	$(1,2)(3,4)$	$-1$
$70$	$3$	$(1,2,3)$	$-1$
$280$	$3$	$(1,2,3)(4,5,6)$	$-1$
$210$	$4$	$(1,2,3,4)$	$-1$
$630$	$4$	$(1,2,3,4)(5,6)$	$1$
$504$	$5$	$(1,2,3,4,5)$	$0$
$210$	$6$	$(1,2,3)(4,5)(6,7)$	$-1$
$420$	$6$	$(1,2,3)(4,5)$	$-1$
$840$	$6$	$(1,2,3,4,5,6)$	$1$
$720$	$7$	$(1,2,3,4,5,6,7)$	$0$
$504$	$10$	$(1,2,3,4,5)(6,7)$	$0$
$420$	$12$	$(1,2,3,4)(5,6,7)$	$-1$

The blue line marks the conjugacy class containing complex conjugation.