Galois orbit of Artin representations 15.7e10_17e8_2017e10.42t411.1

• Label: 15.7e10_17e8_2017e10.42t411.1
• Dimension: 15
• Group: $S_7$
• Conductor: $ 7^{10} \cdot 17^{8} \cdot 2017^{10}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$15$
Group:	$S_7$
Conductor:	$2195991998232776695891479913300120547806284311496241= 7^{10} \cdot 17^{8} \cdot 2017^{10} $
Artin number field:	Splitting field of $f= x^{7} - 2 x^{6} - 2 x^{5} + 6 x^{4} - 3 x^{3} - 2 x^{2} + 4 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 $\Q_{ 191 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 191 }$: $ x^{2} + 190 x + 19 $

Roots:

$r_{ 1 }$	$=$	$ 155 + 112\cdot 191 + 159\cdot 191^{2} + 163\cdot 191^{3} + 65\cdot 191^{4} +O\left(191^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 7 a + 93 + \left(39 a + 139\right)\cdot 191 + \left(44 a + 179\right)\cdot 191^{2} + \left(185 a + 41\right)\cdot 191^{3} + \left(13 a + 140\right)\cdot 191^{4} +O\left(191^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 175 + 160\cdot 191 + 180\cdot 191^{2} + 54\cdot 191^{3} + 42\cdot 191^{4} +O\left(191^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 158 + 126\cdot 191 + 115\cdot 191^{2} + 133\cdot 191^{3} +O\left(191^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 24 a + 126 + 133\cdot 191 + \left(98 a + 113\right)\cdot 191^{2} + \left(99 a + 92\right)\cdot 191^{3} + \left(112 a + 75\right)\cdot 191^{4} +O\left(191^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 184 a + 100 + \left(151 a + 171\right)\cdot 191 + \left(146 a + 184\right)\cdot 191^{2} + \left(5 a + 182\right)\cdot 191^{3} + \left(177 a + 159\right)\cdot 191^{4} +O\left(191^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 167 a + 150 + \left(190 a + 109\right)\cdot 191 + \left(92 a + 20\right)\cdot 191^{2} + \left(91 a + 94\right)\cdot 191^{3} + \left(78 a + 88\right)\cdot 191^{4} +O\left(191^{ 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.