Artin representation 14.89e10_599e10_743e10.42t413.1c1

• Label: 14.89e10_599e10_743e10.42t413.1c1
• Dimension: 14
• Group: $S_7$
• Conductor: $ 89^{10} \cdot 599^{10} \cdot 743^{10}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$14$
Group:	$S_7$
Conductor:	$9507283139058282161693052435435589283724636479431283660851690924154542857649= 89^{10} \cdot 599^{10} \cdot 743^{10} $
Artin number field:	Splitting field of $f= x^{7} - x^{6} - 7 x^{5} + 3 x^{4} + 10 x^{3} - 3 x^{2} - 3 x + 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	42T413
Parity:	Even
Determinant:	1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 137 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 137 }$: $ x^{2} + 131 x + 3 $

Roots:

$r_{ 1 }$	$=$	$ 93 a + 63 + \left(81 a + 73\right)\cdot 137 + \left(78 a + 123\right)\cdot 137^{2} + \left(100 a + 64\right)\cdot 137^{3} + \left(131 a + 94\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 124 a + 136 + \left(85 a + 3\right)\cdot 137 + \left(35 a + 90\right)\cdot 137^{2} + \left(42 a + 1\right)\cdot 137^{3} + \left(107 a + 126\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 56 + 27\cdot 137 + 21\cdot 137^{2} + 129\cdot 137^{3} + 42\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 95 a + 2 + \left(8 a + 84\right)\cdot 137 + \left(100 a + 111\right)\cdot 137^{2} + \left(34 a + 59\right)\cdot 137^{3} + \left(37 a + 45\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 44 a + 73 + \left(55 a + 59\right)\cdot 137 + \left(58 a + 102\right)\cdot 137^{2} + \left(36 a + 41\right)\cdot 137^{3} + \left(5 a + 99\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 42 a + 24 + \left(128 a + 41\right)\cdot 137 + \left(36 a + 18\right)\cdot 137^{2} + \left(102 a + 31\right)\cdot 137^{3} + \left(99 a + 97\right)\cdot 137^{4} +O\left(137^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 13 a + 58 + \left(51 a + 121\right)\cdot 137 + \left(101 a + 80\right)\cdot 137^{2} + \left(94 a + 82\right)\cdot 137^{3} + \left(29 a + 42\right)\cdot 137^{4} +O\left(137^{ 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.