Artin representation 14.11e4_103e4_701e4.21t38.1c1

• Label: 14.11e4_103e4_701e4.21t38.1c1
• Dimension: 14
• Group: $S_7$
• Conductor: $ 11^{4} \cdot 103^{4} \cdot 701^{4}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$14$
Group:	$S_7$
Conductor:	$397916283174105265607521= 11^{4} \cdot 103^{4} \cdot 701^{4} $
Artin number field:	Splitting field of $f= x^{7} - x^{5} - 3 x^{4} - x^{3} + 3 x^{2} + x - 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	21T38
Parity:	Even
Determinant:	1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 113 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 113 }$: $ x^{2} + 101 x + 3 $

Roots:

$r_{ 1 }$	$=$	$ 83 a + 106 + \left(30 a + 15\right)\cdot 113 + \left(78 a + 87\right)\cdot 113^{2} + \left(103 a + 111\right)\cdot 113^{3} + \left(97 a + 96\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 30 a + 85 + \left(82 a + 75\right)\cdot 113 + \left(34 a + 91\right)\cdot 113^{2} + \left(9 a + 34\right)\cdot 113^{3} + \left(15 a + 38\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 74 + 95\cdot 113 + 78\cdot 113^{2} + 72\cdot 113^{3} + 112\cdot 113^{4} +O\left(113^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 85 a + 88 + \left(10 a + 98\right)\cdot 113 + \left(11 a + 57\right)\cdot 113^{2} + \left(79 a + 91\right)\cdot 113^{3} + \left(35 a + 46\right)\cdot 113^{4} +O\left(113^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 61 + 29\cdot 113 + 87\cdot 113^{2} + 94\cdot 113^{3} + 32\cdot 113^{4} +O\left(113^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 60 + 106\cdot 113 + 94\cdot 113^{2} + 33\cdot 113^{3} + 67\cdot 113^{4} +O\left(113^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 28 a + 91 + \left(102 a + 29\right)\cdot 113 + \left(101 a + 67\right)\cdot 113^{2} + \left(33 a + 12\right)\cdot 113^{3} + \left(77 a + 57\right)\cdot 113^{4} +O\left(113^{ 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
$1$	$1$	$()$	$14$
$21$	$2$	$(1,2)$	$6$
$105$	$2$	$(1,2)(3,4)(5,6)$	$2$
$105$	$2$	$(1,2)(3,4)$	$2$
$70$	$3$	$(1,2,3)$	$2$
$280$	$3$	$(1,2,3)(4,5,6)$	$-1$
$210$	$4$	$(1,2,3,4)$	$0$
$630$	$4$	$(1,2,3,4)(5,6)$	$0$
$504$	$5$	$(1,2,3,4,5)$	$-1$
$210$	$6$	$(1,2,3)(4,5)(6,7)$	$2$
$420$	$6$	$(1,2,3)(4,5)$	$0$
$840$	$6$	$(1,2,3,4,5,6)$	$-1$
$720$	$7$	$(1,2,3,4,5,6,7)$	$0$
$504$	$10$	$(1,2,3,4,5)(6,7)$	$1$
$420$	$12$	$(1,2,3,4)(5,6,7)$	$0$

The blue line marks the conjugacy class containing complex conjugation.