Artin representation 35.79e20_97e20_103e20.126.1c1

• Label: 35.79e20_97e20_103e20.126.1c1
• Dimension: 35
• Group: $S_7$
• Conductor: $ 79^{20} \cdot 97^{20} \cdot 103^{20}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$35$
Group:	$S_7$
Conductor:	$8804830608831874330757915695773917774197654835558368994920178550139959779716293617315308404550226271579043936271383201= 79^{20} \cdot 97^{20} \cdot 103^{20} $
Artin number field:	Splitting field of $f= x^{7} - x^{4} - 2 x^{3} + 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 $\Q_{ 127 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 127 }$: $ x^{2} + 126 x + 3 $

Roots:

$r_{ 1 }$	$=$	$ 125 + 104\cdot 127 + 3\cdot 127^{2} + 87\cdot 127^{3} + 73\cdot 127^{4} +O\left(127^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 22 a + 60 + \left(62 a + 105\right)\cdot 127 + \left(11 a + 113\right)\cdot 127^{2} + \left(4 a + 83\right)\cdot 127^{3} + \left(109 a + 117\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 4 a + 34 + \left(107 a + 105\right)\cdot 127 + \left(67 a + 3\right)\cdot 127^{2} + \left(76 a + 124\right)\cdot 127^{3} + \left(123 a + 42\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 31 a + 69 + \left(12 a + 55\right)\cdot 127 + \left(26 a + 45\right)\cdot 127^{2} + \left(39 a + 122\right)\cdot 127^{3} + \left(36 a + 108\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 96 a + 100 + \left(114 a + 36\right)\cdot 127 + \left(100 a + 59\right)\cdot 127^{2} + \left(87 a + 8\right)\cdot 127^{3} + \left(90 a + 106\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 105 a + 82 + \left(64 a + 18\right)\cdot 127 + \left(115 a + 63\right)\cdot 127^{2} + \left(122 a + 76\right)\cdot 127^{3} + \left(17 a + 95\right)\cdot 127^{4} +O\left(127^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 123 a + 38 + \left(19 a + 81\right)\cdot 127 + \left(59 a + 91\right)\cdot 127^{2} + \left(50 a + 5\right)\cdot 127^{3} + \left(3 a + 90\right)\cdot 127^{4} +O\left(127^{ 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.