Galois orbit of Artin representations 15.13e10_53e10_6871e10.42t411.1

• Label: 15.13e10_53e10_6871e10.42t411.1
• Dimension: 15
• Group: $S_7$
• Conductor: $ 13^{10} \cdot 53^{10} \cdot 6871^{10}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$15$
Group:	$S_7$
Conductor:	$5654467399136913451447240695092858482392380172527846687691936346801= 13^{10} \cdot 53^{10} \cdot 6871^{10} $
Artin number field:	Splitting field of $f= x^{7} - x^{6} - 3 x^{5} + 3 x^{4} - x^{2} + 3 x - 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	42T411
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 173 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 173 }$: $ x^{2} + 169 x + 2 $

Roots:

$r_{ 1 }$	$=$	$ 48 a + 164 + \left(89 a + 96\right)\cdot 173 + \left(29 a + 76\right)\cdot 173^{2} + \left(86 a + 34\right)\cdot 173^{3} + \left(48 a + 147\right)\cdot 173^{4} +O\left(173^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 117 + 36\cdot 173 + 73\cdot 173^{2} + 52\cdot 173^{3} + 163\cdot 173^{4} +O\left(173^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 22 + 130\cdot 173 + 103\cdot 173^{2} + 120\cdot 173^{3} + 131\cdot 173^{4} +O\left(173^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 125 a + 10 + \left(83 a + 60\right)\cdot 173 + \left(143 a + 105\right)\cdot 173^{2} + \left(86 a + 3\right)\cdot 173^{3} + \left(124 a + 82\right)\cdot 173^{4} +O\left(173^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 20 + 158\cdot 173 + 160\cdot 173^{2} + 76\cdot 173^{3} + 136\cdot 173^{4} +O\left(173^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 134 + 134\cdot 173 + 22\cdot 173^{2} + 41\cdot 173^{3} + 97\cdot 173^{4} +O\left(173^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 53 + 75\cdot 173 + 149\cdot 173^{2} + 16\cdot 173^{3} + 107\cdot 173^{4} +O\left(173^{ 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
			$c1$
$1$	$1$	$()$	$15$
$21$	$2$	$(1,2)$	$-5$
$105$	$2$	$(1,2)(3,4)(5,6)$	$3$
$105$	$2$	$(1,2)(3,4)$	$-1$
$70$	$3$	$(1,2,3)$	$3$
$280$	$3$	$(1,2,3)(4,5,6)$	$0$
$210$	$4$	$(1,2,3,4)$	$-1$
$630$	$4$	$(1,2,3,4)(5,6)$	$-1$
$504$	$5$	$(1,2,3,4,5)$	$0$
$210$	$6$	$(1,2,3)(4,5)(6,7)$	$-1$
$420$	$6$	$(1,2,3)(4,5)$	$1$
$840$	$6$	$(1,2,3,4,5,6)$	$0$
$720$	$7$	$(1,2,3,4,5,6,7)$	$1$
$504$	$10$	$(1,2,3,4,5)(6,7)$	$0$
$420$	$12$	$(1,2,3,4)(5,6,7)$	$-1$

The blue line marks the conjugacy class containing complex conjugation.