Galois orbit of Artin representations 21.197e11_211e11_1031e11.42t418.1

• Label: 21.197e11_211e11_1031e11.42t418.1
• Dimension: 21
• Group: $S_7$
• Conductor: $ 197^{11} \cdot 211^{11} \cdot 1031^{11}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$21$
Group:	$S_7$
Conductor:	$895531391875285848449631228739479603770147280932916240179734825437001442357616614473= 197^{11} \cdot 211^{11} \cdot 1031^{11} $
Artin number field:	Splitting field of $f= x^{7} - x^{6} - 7 x^{5} + 3 x^{4} + 14 x^{3} + 2 x^{2} - 4 x - 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	42T418
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 101 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 101 }$: $ x^{2} + 97 x + 2 $

Roots:

$r_{ 1 }$	$=$	$ 87 a + 83 + \left(86 a + 66\right)\cdot 101 + \left(3 a + 57\right)\cdot 101^{2} + \left(24 a + 3\right)\cdot 101^{3} + \left(69 a + 11\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 27 a + 28 + \left(17 a + 78\right)\cdot 101 + \left(34 a + 85\right)\cdot 101^{2} + \left(28 a + 96\right)\cdot 101^{3} + \left(86 a + 32\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 14 a + 27 + \left(14 a + 24\right)\cdot 101 + \left(97 a + 87\right)\cdot 101^{2} + \left(76 a + 95\right)\cdot 101^{3} + \left(31 a + 61\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 74 a + 35 + \left(83 a + 19\right)\cdot 101 + \left(66 a + 3\right)\cdot 101^{2} + \left(72 a + 75\right)\cdot 101^{3} + \left(14 a + 46\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 9 + 89\cdot 101 + 55\cdot 101^{2} + 51\cdot 101^{3} + 73\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 71 + 70\cdot 101 + 80\cdot 101^{2} + 67\cdot 101^{3} + 52\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 51 + 55\cdot 101 + 33\cdot 101^{2} + 13\cdot 101^{3} + 24\cdot 101^{4} +O\left(101^{ 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.