Galois orbit of Artin representations 14.23e10_31e10_5711e10.42t413.1

• Label: 14.23e10_31e10_5711e10.42t413.1
• Dimension: 14
• Group: $S_7$
• Conductor: $ 23^{10} \cdot 31^{10} \cdot 5711^{10}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$14$
Group:	$S_7$
Conductor:	$1253189940715098530611652882160842138931741001453214939925935851249= 23^{10} \cdot 31^{10} \cdot 5711^{10} $
Artin number field:	Splitting field of $f= x^{7} - x^{6} - 3 x^{5} + 2 x^{4} + x^{3} + x^{2} + x - 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	42T413
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 103 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 103 }$: $ x^{2} + 102 x + 5 $

Roots:

$r_{ 1 }$	$=$	$ a + 71 + \left(63 a + 75\right)\cdot 103 + 58 a\cdot 103^{2} + \left(58 a + 68\right)\cdot 103^{3} + \left(97 a + 25\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 65 a + 48 + \left(8 a + 64\right)\cdot 103 + \left(96 a + 77\right)\cdot 103^{2} + \left(73 a + 19\right)\cdot 103^{3} + \left(49 a + 41\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 79 + 54\cdot 103 + 95\cdot 103^{2} + 40\cdot 103^{3} + 96\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 38 a + 10 + \left(94 a + 8\right)\cdot 103 + \left(6 a + 62\right)\cdot 103^{2} + \left(29 a + 100\right)\cdot 103^{3} + \left(53 a + 16\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 83 + 93\cdot 103 + 86\cdot 103^{2} + 41\cdot 103^{3} + 45\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 102 a + 72 + \left(39 a + 34\right)\cdot 103 + \left(44 a + 99\right)\cdot 103^{2} + \left(44 a + 67\right)\cdot 103^{3} + \left(5 a + 64\right)\cdot 103^{4} +O\left(103^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 50 + 80\cdot 103 + 92\cdot 103^{2} + 72\cdot 103^{3} + 18\cdot 103^{4} +O\left(103^{ 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 values
			$c1$
$1$	$1$	$()$	$14$
$21$	$2$	$(1,2)$	$-6$
$105$	$2$	$(1,2)(3,4)(5,6)$	$-2$
$105$	$2$	$(1,2)(3,4)$	$2$
$70$	$3$	$(1,2,3)$	$2$
$280$	$3$	$(1,2,3)(4,5,6)$	$-1$
$210$	$4$	$(1,2,3,4)$	$0$
$630$	$4$	$(1,2,3,4)(5,6)$	$0$
$504$	$5$	$(1,2,3,4,5)$	$-1$
$210$	$6$	$(1,2,3)(4,5)(6,7)$	$2$
$420$	$6$	$(1,2,3)(4,5)$	$0$
$840$	$6$	$(1,2,3,4,5,6)$	$1$
$720$	$7$	$(1,2,3,4,5,6,7)$	$0$
$504$	$10$	$(1,2,3,4,5)(6,7)$	$-1$
$420$	$12$	$(1,2,3,4)(5,6,7)$	$0$

The blue line marks the conjugacy class containing complex conjugation.