Artin representation 35.4520279e15.70.1c1

• Label: 35.4520279e15.70.1c1
• Dimension: 35
• Group: $S_7$
• Conductor: $ 4520279^{15}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$35$
Group:	$S_7$
Conductor:	$6721692124623812707800147970045429473929869923509138979528218089170057372173979508026972997327732199= 4520279^{15} $
Artin number field:	Splitting field of $f= x^{7} - x^{6} - x^{5} + 3 x^{4} - 4 x^{3} - 2 x^{2} + 4 x - 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	70
Parity:	Odd
Determinant:	1.4520279.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 131 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 131 }$: $ x^{2} + 127 x + 2 $

Roots:

$r_{ 1 }$	$=$	$ 88 a + 47 + \left(126 a + 8\right)\cdot 131 + \left(76 a + 46\right)\cdot 131^{2} + \left(83 a + 40\right)\cdot 131^{3} + \left(77 a + 32\right)\cdot 131^{4} +O\left(131^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 3 a + 10 + \left(52 a + 85\right)\cdot 131 + \left(33 a + 105\right)\cdot 131^{2} + \left(97 a + 125\right)\cdot 131^{3} + \left(2 a + 71\right)\cdot 131^{4} +O\left(131^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 43 a + 6 + \left(4 a + 34\right)\cdot 131 + \left(54 a + 96\right)\cdot 131^{2} + \left(47 a + 35\right)\cdot 131^{3} + \left(53 a + 128\right)\cdot 131^{4} +O\left(131^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 39 + 40\cdot 131 + 92\cdot 131^{2} + 56\cdot 131^{3} + 46\cdot 131^{4} +O\left(131^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 50 + 50\cdot 131 + 29\cdot 131^{2} + 110\cdot 131^{3} + 119\cdot 131^{4} +O\left(131^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 89 + 15\cdot 131 + 98\cdot 131^{2} + 66\cdot 131^{3} + 8\cdot 131^{4} +O\left(131^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 128 a + 22 + \left(78 a + 28\right)\cdot 131 + \left(97 a + 56\right)\cdot 131^{2} + \left(33 a + 88\right)\cdot 131^{3} + \left(128 a + 116\right)\cdot 131^{4} +O\left(131^{ 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.