Artin representation 14.11e10_309493e10.42t413.1c1

• Label: 14.11e10_309493e10.42t413.1c1
• Dimension: 14
• Group: $S_7$
• Conductor: $ 11^{10} \cdot 309493^{10}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$14$
Group:	$S_7$
Conductor:	$209139063568253827185991688686505717861850854605158991762324785649= 11^{10} \cdot 309493^{10} $
Artin number field:	Splitting field of $f= x^{7} - 2 x^{5} - 3 x^{4} + x^{3} + 5 x^{2} - 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 $\Q_{ 43 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $ x^{2} + 42 x + 3 $

Roots:

$r_{ 1 }$	$=$	$ a + 15 + \left(2 a + 2\right)\cdot 43 + \left(13 a + 8\right)\cdot 43^{2} + \left(11 a + 10\right)\cdot 43^{3} + \left(9 a + 21\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 12 a + 16 + \left(2 a + 37\right)\cdot 43 + \left(9 a + 13\right)\cdot 43^{2} + 43^{3} + \left(4 a + 34\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 42 a + 16 + \left(40 a + 3\right)\cdot 43 + \left(29 a + 19\right)\cdot 43^{2} + \left(31 a + 8\right)\cdot 43^{3} + \left(33 a + 19\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 23 + 15\cdot 43 + 35\cdot 43^{2} + 33\cdot 43^{3} + 38\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 2 + 19\cdot 43 + 21\cdot 43^{2} + 23\cdot 43^{3} + 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 31 a + 28 + \left(40 a + 27\right)\cdot 43 + \left(33 a + 20\right)\cdot 43^{2} + \left(42 a + 35\right)\cdot 43^{3} + \left(38 a + 37\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 29 + 23\cdot 43 + 10\cdot 43^{2} + 16\cdot 43^{3} + 19\cdot 43^{4} +O\left(43^{ 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.