Galois orbit of Artin representations 35.5e20_13e20_1205753e20.126.1

• Label: 35.5e20_13e20_1205753e20.126.1
• Dimension: 35
• Group: $S_7$
• Conductor: $ 5^{20} \cdot 13^{20} \cdot 1205753^{20}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$35$
Group:	$S_7$
Conductor:	$76459975738863950321023156593824192340765497217279491332886235304950511216380272131875038226105311895163130854426578540284431886750764345385856723785400390625= 5^{20} \cdot 13^{20} \cdot 1205753^{20} $
Artin number field:	Splitting field of $f= x^{7} - x^{6} - 7 x^{5} + 3 x^{4} + 13 x^{3} + x^{2} - 4 x - 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	126
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 211 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 211 }$: $ x^{2} + 207 x + 2 $

Roots:

$r_{ 1 }$	$=$	$ 185 a + 115 + \left(147 a + 159\right)\cdot 211 + \left(157 a + 90\right)\cdot 211^{2} + \left(134 a + 22\right)\cdot 211^{3} + \left(91 a + 49\right)\cdot 211^{4} +O\left(211^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 35 + 122\cdot 211 + 141\cdot 211^{2} + 115\cdot 211^{3} + 183\cdot 211^{4} +O\left(211^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 26 a + 11 + \left(63 a + 144\right)\cdot 211 + \left(53 a + 151\right)\cdot 211^{2} + \left(76 a + 192\right)\cdot 211^{3} + \left(119 a + 69\right)\cdot 211^{4} +O\left(211^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 201 a + 132 + \left(84 a + 8\right)\cdot 211 + \left(27 a + 80\right)\cdot 211^{2} + 102 a\cdot 211^{3} + \left(172 a + 25\right)\cdot 211^{4} +O\left(211^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 195 + 165\cdot 211 + 176\cdot 211^{2} + 142\cdot 211^{3} + 202\cdot 211^{4} +O\left(211^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 54 + 96\cdot 211 + 98\cdot 211^{2} + 199\cdot 211^{3} + 122\cdot 211^{4} +O\left(211^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 10 a + 92 + \left(126 a + 147\right)\cdot 211 + \left(183 a + 104\right)\cdot 211^{2} + \left(108 a + 170\right)\cdot 211^{3} + \left(38 a + 190\right)\cdot 211^{4} +O\left(211^{ 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
			$c1$
$1$	$1$	$()$	$35$
$21$	$2$	$(1,2)$	$-5$
$105$	$2$	$(1,2)(3,4)(5,6)$	$-1$
$105$	$2$	$(1,2)(3,4)$	$-1$
$70$	$3$	$(1,2,3)$	$-1$
$280$	$3$	$(1,2,3)(4,5,6)$	$-1$
$210$	$4$	$(1,2,3,4)$	$1$
$630$	$4$	$(1,2,3,4)(5,6)$	$1$
$504$	$5$	$(1,2,3,4,5)$	$0$
$210$	$6$	$(1,2,3)(4,5)(6,7)$	$-1$
$420$	$6$	$(1,2,3)(4,5)$	$1$
$840$	$6$	$(1,2,3,4,5,6)$	$-1$
$720$	$7$	$(1,2,3,4,5,6,7)$	$0$
$504$	$10$	$(1,2,3,4,5)(6,7)$	$0$
$420$	$12$	$(1,2,3,4)(5,6,7)$	$1$

The blue line marks the conjugacy class containing complex conjugation.