Galois orbit of Artin representations 21.19e10_83e10_577e10.84.1

• Label: 21.19e10_83e10_577e10.84.1
• Dimension: 21
• Group: $S_7$
• Conductor: $ 19^{10} \cdot 83^{10} \cdot 577^{10}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$21$
Group:	$S_7$
Conductor:	$389112394612258426837336537919063029088953395046449327781201= 19^{10} \cdot 83^{10} \cdot 577^{10} $
Artin number field:	Splitting field of $f= x^{7} - 2 x^{5} - x^{4} + 4 x^{3} + 2 x^{2} - 2 x - 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	84
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 97 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$: $ x^{2} + 96 x + 5 $

Roots:

$r_{ 1 }$	$=$	$ 28 + 81\cdot 97 + 93\cdot 97^{2} + 12\cdot 97^{3} + 59\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 64 + 50\cdot 97 + 7\cdot 97^{2} + 83\cdot 97^{3} + 65\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 29 + 26\cdot 97 + 28\cdot 97^{2} + 33\cdot 97^{3} + 86\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 32 a + 85 + \left(29 a + 7\right)\cdot 97 + \left(92 a + 27\right)\cdot 97^{2} + \left(2 a + 83\right)\cdot 97^{3} + \left(52 a + 68\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 65 a + 20 + \left(67 a + 5\right)\cdot 97 + \left(4 a + 90\right)\cdot 97^{2} + \left(94 a + 90\right)\cdot 97^{3} + \left(44 a + 20\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 75 + 27\cdot 97 + 25\cdot 97^{2} + 60\cdot 97^{3} + 29\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 87 + 91\cdot 97 + 18\cdot 97^{2} + 24\cdot 97^{3} + 57\cdot 97^{4} +O\left(97^{ 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
			$c1$
$1$	$1$	$()$	$21$
$21$	$2$	$(1,2)$	$1$
$105$	$2$	$(1,2)(3,4)(5,6)$	$-3$
$105$	$2$	$(1,2)(3,4)$	$1$
$70$	$3$	$(1,2,3)$	$-3$
$280$	$3$	$(1,2,3)(4,5,6)$	$0$
$210$	$4$	$(1,2,3,4)$	$-1$
$630$	$4$	$(1,2,3,4)(5,6)$	$-1$
$504$	$5$	$(1,2,3,4,5)$	$1$
$210$	$6$	$(1,2,3)(4,5)(6,7)$	$1$
$420$	$6$	$(1,2,3)(4,5)$	$1$
$840$	$6$	$(1,2,3,4,5,6)$	$0$
$720$	$7$	$(1,2,3,4,5,6,7)$	$0$
$504$	$10$	$(1,2,3,4,5)(6,7)$	$1$
$420$	$12$	$(1,2,3,4)(5,6,7)$	$-1$

The blue line marks the conjugacy class containing complex conjugation.