Galois orbit of Artin representations 21.571e10_1699e10.84.1

• Label: 21.571e10_1699e10.84.1
• Dimension: 21
• Group: $S_7$
• Conductor: $ 571^{10} \cdot 1699^{10}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$21$
Group:	$S_7$
Conductor:	$738405412071075642174022520000845062962106329648881080919201= 571^{10} \cdot 1699^{10} $
Artin number field:	Splitting field of $f= x^{7} - x^{6} - 2 x^{5} + 2 x^{4} + x^{3} - 2 x^{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 $\Q_{ 67 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: $ x^{2} + 63 x + 2 $

Roots:

$r_{ 1 }$	$=$	$ 18 + 22\cdot 67 + 26\cdot 67^{2} + 28\cdot 67^{3} + 42\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 52 + 6\cdot 67 + 58\cdot 67^{2} + 35\cdot 67^{3} + 60\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 48 a + 3 + \left(60 a + 14\right)\cdot 67 + \left(63 a + 32\right)\cdot 67^{2} + \left(22 a + 14\right)\cdot 67^{3} + \left(38 a + 56\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 19 a + 61 + \left(6 a + 7\right)\cdot 67 + \left(3 a + 26\right)\cdot 67^{2} + \left(44 a + 42\right)\cdot 67^{3} + \left(28 a + 52\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 51 + 27\cdot 67 + 52\cdot 67^{2} + 48\cdot 67^{3} + 23\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 16 a + 10 + \left(65 a + 39\right)\cdot 67 + \left(26 a + 48\right)\cdot 67^{2} + \left(17 a + 27\right)\cdot 67^{3} + \left(44 a + 3\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 51 a + 7 + \left(a + 16\right)\cdot 67 + \left(40 a + 24\right)\cdot 67^{2} + \left(49 a + 3\right)\cdot 67^{3} + \left(22 a + 29\right)\cdot 67^{4} +O\left(67^{ 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.