Artin representation 14.7e10_353e10_2137e10.42t413.1c1

• Label: 14.7e10_353e10_2137e10.42t413.1c1
• Dimension: 14
• Group: $S_7$
• Conductor: $ 7^{10} \cdot 353^{10} \cdot 2137^{10}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$14$
Group:	$S_7$
Conductor:	$16856704578141183965784213061429615337905846220316711729994881879649= 7^{10} \cdot 353^{10} \cdot 2137^{10} $
Artin number field:	Splitting field of $f= x^{7} - x^{5} - 3 x^{4} - 3 x^{3} + 5 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 $\Q_{ 107 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 107 }$: $ x^{2} + 103 x + 2 $

Roots:

$r_{ 1 }$	$=$	$ 69 a + 96 + \left(38 a + 104\right)\cdot 107 + \left(41 a + 19\right)\cdot 107^{2} + \left(93 a + 12\right)\cdot 107^{3} + \left(26 a + 96\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 38 a + 51 + \left(68 a + 83\right)\cdot 107 + \left(65 a + 39\right)\cdot 107^{2} + \left(13 a + 23\right)\cdot 107^{3} + \left(80 a + 3\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 105 + 43\cdot 107 + 42\cdot 107^{2} + 84\cdot 107^{3} + 59\cdot 107^{4} +O\left(107^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 75 + 50\cdot 107 + 40\cdot 107^{2} + 35\cdot 107^{3} + 7\cdot 107^{4} +O\left(107^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 6 + 43\cdot 107 + 89\cdot 107^{2} + 60\cdot 107^{3} + 39\cdot 107^{4} +O\left(107^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 65 a + 78 + \left(78 a + 86\right)\cdot 107 + \left(a + 26\right)\cdot 107^{2} + \left(89 a + 89\right)\cdot 107^{3} + \left(10 a + 26\right)\cdot 107^{4} +O\left(107^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 42 a + 17 + \left(28 a + 15\right)\cdot 107 + \left(105 a + 62\right)\cdot 107^{2} + \left(17 a + 15\right)\cdot 107^{3} + \left(96 a + 88\right)\cdot 107^{4} +O\left(107^{ 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.