Galois orbit of Artin representations 20.3359e10_19559e10.70.1

• Label: 20.3359e10_19559e10.70.1
• Dimension: 20
• Group: $S_7$
• Conductor: $ 3359^{10} \cdot 19559^{10}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$20$
Group:	$S_7$
Conductor:	$1498188659314490853204858202731600676455532583276606206803104552707437662434801= 3359^{10} \cdot 19559^{10} $
Artin number field:	Splitting field of $f= x^{7} - x^{6} - 8 x^{5} + x^{4} + 12 x^{3} - x^{2} - 4 x + 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	70
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 197 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 197 }$: $ x^{2} + 192 x + 2 $

Roots:

$r_{ 1 }$	$=$	$ 146 + 125\cdot 197 + 105\cdot 197^{2} + 138\cdot 197^{3} + 42\cdot 197^{4} +O\left(197^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 157 a + 172 + \left(73 a + 8\right)\cdot 197 + \left(32 a + 55\right)\cdot 197^{2} + \left(76 a + 195\right)\cdot 197^{3} + \left(3 a + 44\right)\cdot 197^{4} +O\left(197^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 160 + 167\cdot 197 + 22\cdot 197^{2} + 12\cdot 197^{3} + 73\cdot 197^{4} +O\left(197^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 98 + 105\cdot 197 + 73\cdot 197^{2} + 145\cdot 197^{3} + 126\cdot 197^{4} +O\left(197^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 165 + 85\cdot 197 + 22\cdot 197^{2} + 57\cdot 197^{3} + 50\cdot 197^{4} +O\left(197^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 40 a + 169 + \left(123 a + 23\right)\cdot 197 + \left(164 a + 143\right)\cdot 197^{2} + \left(120 a + 149\right)\cdot 197^{3} + \left(193 a + 182\right)\cdot 197^{4} +O\left(197^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 76 + 73\cdot 197 + 168\cdot 197^{2} + 89\cdot 197^{3} + 70\cdot 197^{4} +O\left(197^{ 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
			$c1$
$1$	$1$	$()$	$20$
$21$	$2$	$(1,2)$	$0$
$105$	$2$	$(1,2)(3,4)(5,6)$	$0$
$105$	$2$	$(1,2)(3,4)$	$-4$
$70$	$3$	$(1,2,3)$	$2$
$280$	$3$	$(1,2,3)(4,5,6)$	$2$
$210$	$4$	$(1,2,3,4)$	$0$
$630$	$4$	$(1,2,3,4)(5,6)$	$0$
$504$	$5$	$(1,2,3,4,5)$	$0$
$210$	$6$	$(1,2,3)(4,5)(6,7)$	$2$
$420$	$6$	$(1,2,3)(4,5)$	$0$
$840$	$6$	$(1,2,3,4,5,6)$	$0$
$720$	$7$	$(1,2,3,4,5,6,7)$	$-1$
$504$	$10$	$(1,2,3,4,5)(6,7)$	$0$
$420$	$12$	$(1,2,3,4)(5,6,7)$	$0$

The blue line marks the conjugacy class containing complex conjugation.