Galois orbit of Artin representations 21.1423e10_2153e10.84.1

• Label: 21.1423e10_2153e10.84.1
• Dimension: 21
• Group: $S_7$
• Conductor: $ 1423^{10} \cdot 2153^{10}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$21$
Group:	$S_7$
Conductor:	$72860022173986860085328861712672467445384199288317198044549530801= 1423^{10} \cdot 2153^{10} $
Artin number field:	Splitting field of $f= x^{7} - 3 x^{6} + x^{5} + 4 x^{4} - 5 x^{3} + x^{2} + 3 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 $\Q_{ 47 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{2} + 45 x + 5 $

Roots:

$r_{ 1 }$	$=$	$ 25 + 45\cdot 47 + 44\cdot 47^{2} + 47^{3} + 44\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 24 a + 43 + \left(19 a + 11\right)\cdot 47 + \left(27 a + 25\right)\cdot 47^{2} + \left(3 a + 26\right)\cdot 47^{3} + \left(37 a + 37\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 24 + 27\cdot 47 + 16\cdot 47^{2} + 23\cdot 47^{3} + 29\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 23 a + 44 + \left(27 a + 26\right)\cdot 47 + \left(19 a + 13\right)\cdot 47^{2} + \left(43 a + 6\right)\cdot 47^{3} + \left(9 a + 14\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 26 + 4\cdot 47 + 10\cdot 47^{2} + 46\cdot 47^{3} + 22\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 20 + 35\cdot 47^{2} + 14\cdot 47^{3} + 22\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 9 + 24\cdot 47 + 42\cdot 47^{2} + 21\cdot 47^{3} + 17\cdot 47^{4} +O\left(47^{ 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.