Artin representation 14.137e10_7937e10.42t413.1c1

• Label: 14.137e10_7937e10.42t413.1c1
• Dimension: 14
• Group: $S_7$
• Conductor: $ 137^{10} \cdot 7937^{10}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$14$
Group:	$S_7$
Conductor:	$2310837861895363664655878241212723505100831601194310062726801= 137^{10} \cdot 7937^{10} $
Artin number field:	Splitting field of $f= x^{7} - x^{6} - x^{5} + 3 x^{4} - 2 x^{3} - x^{2} + 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 $\Q_{ 109 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 109 }$: $ x^{2} + 108 x + 6 $

Roots:

$r_{ 1 }$	$=$	$ 83 a + 80 + \left(19 a + 76\right)\cdot 109 + \left(53 a + 64\right)\cdot 109^{2} + \left(75 a + 99\right)\cdot 109^{3} + \left(58 a + 21\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 79 + 72\cdot 109 + 101\cdot 109^{2} + 77\cdot 109^{3} + 6\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 66 a + 66 + \left(8 a + 83\right)\cdot 109 + \left(14 a + 91\right)\cdot 109^{2} + \left(57 a + 25\right)\cdot 109^{3} + \left(108 a + 64\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 26 a + 54 + \left(89 a + 13\right)\cdot 109 + \left(55 a + 98\right)\cdot 109^{2} + \left(33 a + 12\right)\cdot 109^{3} + \left(50 a + 5\right)\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 55 + 30\cdot 109 + 101\cdot 109^{2} + 104\cdot 109^{3} + 26\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 80 + 23\cdot 109 + 99\cdot 109^{2} + 45\cdot 109^{3} + 86\cdot 109^{4} +O\left(109^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 43 a + 23 + \left(100 a + 26\right)\cdot 109 + \left(94 a + 97\right)\cdot 109^{2} + \left(51 a + 68\right)\cdot 109^{3} + 6\cdot 109^{4} +O\left(109^{ 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.