Artin representation 35.317e15_8803e15.70.1c1

• Label: 35.317e15_8803e15.70.1c1
• Dimension: 35
• Group: $S_7$
• Conductor: $ 317^{15} \cdot 8803^{15}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$35$
Group:	$S_7$
Conductor:	$4845621204197274365006154815892672338694417593716836098393301459529346347737521687538692612995751= 317^{15} \cdot 8803^{15} $
Artin number field:	Splitting field of $f= x^{7} - 2 x^{5} - 3 x^{3} - x^{2} + 3 x + 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	70
Parity:	Odd
Determinant:	1.317_8803.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 151 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 151 }$: $ x^{2} + 149 x + 6 $

Roots:

$r_{ 1 }$	$=$	$ 107 + 97\cdot 151 + 91\cdot 151^{2} + 53\cdot 151^{3} + 140\cdot 151^{4} +O\left(151^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 27 a + 120 + \left(82 a + 19\right)\cdot 151 + \left(78 a + 90\right)\cdot 151^{2} + \left(41 a + 95\right)\cdot 151^{3} + \left(15 a + 45\right)\cdot 151^{4} +O\left(151^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 58 + 140\cdot 151 + 89\cdot 151^{2} + 46\cdot 151^{3} + 62\cdot 151^{4} +O\left(151^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 64 a + 69 + \left(150 a + 5\right)\cdot 151 + \left(46 a + 71\right)\cdot 151^{2} + 55 a\cdot 151^{3} + \left(133 a + 53\right)\cdot 151^{4} +O\left(151^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 30 + 92\cdot 151 + 81\cdot 151^{2} + 92\cdot 151^{3} + 3\cdot 151^{4} +O\left(151^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 124 a + 23 + \left(68 a + 6\right)\cdot 151 + \left(72 a + 14\right)\cdot 151^{2} + \left(109 a + 100\right)\cdot 151^{3} + \left(135 a + 34\right)\cdot 151^{4} +O\left(151^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 87 a + 46 + 91\cdot 151 + \left(104 a + 14\right)\cdot 151^{2} + \left(95 a + 64\right)\cdot 151^{3} + \left(17 a + 113\right)\cdot 151^{4} +O\left(151^{ 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.