Artin representation 35.3e18_43e20_173539e20.126.1c1

• Label: 35.3e18_43e20_173539e20.126.1c1
• Dimension: 35
• Group: $S_7$
• Conductor: $ 3^{18} \cdot 43^{20} \cdot 173539^{20}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$35$
Group:	$S_7$
Conductor:	$111043593468939894777832433884616815573775628869930955564034437245830597052224501575779220427796669368137935695444000253508705667186454730717881289= 3^{18} \cdot 43^{20} \cdot 173539^{20} $
Artin number field:	Splitting field of $f= x^{7} - 3 x^{6} - 3 x^{5} + 11 x^{4} + 2 x^{3} - 10 x^{2} + 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	126
Parity:	Even
Determinant:	1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 83 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 83 }$: $ x^{2} + 82 x + 2 $

Roots:

$r_{ 1 }$	$=$	$ 12 a + 55 + \left(80 a + 50\right)\cdot 83 + \left(8 a + 1\right)\cdot 83^{2} + \left(21 a + 53\right)\cdot 83^{3} + \left(17 a + 13\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 75 a + 22 + \left(17 a + 34\right)\cdot 83 + \left(81 a + 32\right)\cdot 83^{2} + \left(15 a + 43\right)\cdot 83^{3} + \left(55 a + 52\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 23 a + 58 + \left(15 a + 45\right)\cdot 83 + \left(66 a + 53\right)\cdot 83^{2} + \left(33 a + 76\right)\cdot 83^{3} + \left(5 a + 33\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 38 + 67\cdot 83 + 30\cdot 83^{2} + 71\cdot 83^{3} + 41\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 60 a + 81 + \left(67 a + 37\right)\cdot 83 + \left(16 a + 21\right)\cdot 83^{2} + \left(49 a + 44\right)\cdot 83^{3} + \left(77 a + 5\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 71 a + 67 + \left(2 a + 35\right)\cdot 83 + \left(74 a + 13\right)\cdot 83^{2} + \left(61 a + 65\right)\cdot 83^{3} + \left(65 a + 9\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 8 a + 14 + \left(65 a + 60\right)\cdot 83 + \left(a + 12\right)\cdot 83^{2} + \left(67 a + 61\right)\cdot 83^{3} + \left(27 a + 8\right)\cdot 83^{4} +O\left(83^{ 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.