Artin representation 6.2e6_728809.7t7.1c1

• Label: 6.2e6_728809.7t7.1c1
• Dimension: 6
• Group: $S_7$
• Conductor: $ 2^{6} \cdot 728809 $
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$6$
Group:	$S_7$
Conductor:	$46643776= 2^{6} \cdot 728809 $
Artin number field:	Splitting field of $f= x^{7} - x^{6} - 7 x^{5} + 3 x^{4} + 13 x^{3} - x^{2} - 5 x - 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$S_7$
Parity:	Even
Determinant:	1.728809.2t1.1c1

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:

$r_{ 1 }$	$=$	$ 54 + 53\cdot 191 + 22\cdot 191^{2} + 92\cdot 191^{3} + 174\cdot 191^{4} +O\left(191^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 68 a + 138 + \left(180 a + 122\right)\cdot 191 + \left(56 a + 47\right)\cdot 191^{2} + \left(118 a + 154\right)\cdot 191^{3} + \left(60 a + 100\right)\cdot 191^{4} +O\left(191^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 123 a + 15 + \left(10 a + 44\right)\cdot 191 + \left(134 a + 115\right)\cdot 191^{2} + \left(72 a + 24\right)\cdot 191^{3} + \left(130 a + 43\right)\cdot 191^{4} +O\left(191^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 98 a + 15 + \left(20 a + 35\right)\cdot 191 + \left(153 a + 143\right)\cdot 191^{2} + \left(154 a + 63\right)\cdot 191^{3} + \left(49 a + 161\right)\cdot 191^{4} +O\left(191^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 117 + 116\cdot 191 + 101\cdot 191^{2} + 149\cdot 191^{3} + 149\cdot 191^{4} +O\left(191^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 122 + 52\cdot 191 + 58\cdot 191^{2} + 23\cdot 191^{3} + 78\cdot 191^{4} +O\left(191^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 93 a + 113 + \left(170 a + 148\right)\cdot 191 + \left(37 a + 84\right)\cdot 191^{2} + \left(36 a + 65\right)\cdot 191^{3} + \left(141 a + 56\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 value
$1$	$1$	$()$	$6$
$21$	$2$	$(1,2)$	$4$
$105$	$2$	$(1,2)(3,4)(5,6)$	$0$
$105$	$2$	$(1,2)(3,4)$	$2$
$70$	$3$	$(1,2,3)$	$3$
$280$	$3$	$(1,2,3)(4,5,6)$	$0$
$210$	$4$	$(1,2,3,4)$	$2$
$630$	$4$	$(1,2,3,4)(5,6)$	$0$
$504$	$5$	$(1,2,3,4,5)$	$1$
$210$	$6$	$(1,2,3)(4,5)(6,7)$	$-1$
$420$	$6$	$(1,2,3)(4,5)$	$1$
$840$	$6$	$(1,2,3,4,5,6)$	$0$
$720$	$7$	$(1,2,3,4,5,6,7)$	$-1$
$504$	$10$	$(1,2,3,4,5)(6,7)$	$-1$
$420$	$12$	$(1,2,3,4)(5,6,7)$	$-1$

The blue line marks the conjugacy class containing complex conjugation.