Galois orbit of Artin representations 35.11e20_97e20_373e20.126.1

• Label: 35.11e20_97e20_373e20.126.1
• Dimension: 35
• Group: $S_7$
• Conductor: $ 11^{20} \cdot 97^{20} \cdot 373^{20}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$35$
Group:	$S_7$
Conductor:	$9941800057452612745930045935410067053220169830449051243589740224768289635948667949143195881805293714669988488801= 11^{20} \cdot 97^{20} \cdot 373^{20} $
Artin number field:	Splitting field of $f= x^{7} - 2 x^{5} + 3 x^{3} - x^{2} - 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 $\Q_{ 83 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 83 }$: $ x^{2} + 82 x + 2 $

Roots:

$r_{ 1 }$	$=$	$ 2 + 32\cdot 83 + 6\cdot 83^{2} + 41\cdot 83^{3} + 26\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 37 + 31\cdot 83 + 6\cdot 83^{2} + 37\cdot 83^{3} + 10\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 79 a + 76 + 32\cdot 83 + \left(52 a + 63\right)\cdot 83^{2} + \left(27 a + 23\right)\cdot 83^{3} + \left(63 a + 16\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 4 a + 72 + \left(82 a + 37\right)\cdot 83 + \left(30 a + 31\right)\cdot 83^{2} + \left(55 a + 82\right)\cdot 83^{3} + \left(19 a + 51\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 8 a + 76 + \left(51 a + 65\right)\cdot 83 + \left(45 a + 24\right)\cdot 83^{2} + \left(52 a + 71\right)\cdot 83^{3} + \left(18 a + 4\right)\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 68 + 22\cdot 83 + 14\cdot 83^{2} + 81\cdot 83^{3} + 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 75 a + 1 + \left(31 a + 26\right)\cdot 83 + \left(37 a + 19\right)\cdot 83^{2} + \left(30 a + 78\right)\cdot 83^{3} + \left(64 a + 53\right)\cdot 83^{4} +O\left(83^{ 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.