Galois orbit of Artin representations 35.2e58_101e20_26759e20.126.1

• Label: 35.2e58_101e20_26759e20.126.1
• Dimension: 35
• Group: $S_7$
• Conductor: $ 2^{58} \cdot 101^{20} \cdot 26759^{20}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$35$
Group:	$S_7$
Conductor:	$124613419959355375360834110556242539211541016356972991438083476210535005333701008256975073344266765923944608054835380620018500556580991210245586944= 2^{58} \cdot 101^{20} \cdot 26759^{20} $
Artin number field:	Splitting field of $f= x^{7} - x^{6} - 7 x^{5} + 4 x^{4} + 15 x^{3} - 2 x^{2} - 8 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 $\Q_{ 59 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $ x^{2} + 58 x + 2 $

Roots:

$r_{ 1 }$	$=$	$ 50 + 40\cdot 59 + 32\cdot 59^{2} + 49\cdot 59^{3} + 32\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 27 a + 8 + \left(17 a + 12\right)\cdot 59 + \left(14 a + 1\right)\cdot 59^{2} + \left(2 a + 5\right)\cdot 59^{3} + \left(50 a + 35\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 46 + 12\cdot 59 + 19\cdot 59^{2} + 55\cdot 59^{3} + 7\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 32 a + 35 + \left(41 a + 2\right)\cdot 59 + \left(44 a + 57\right)\cdot 59^{2} + \left(56 a + 51\right)\cdot 59^{3} + \left(8 a + 23\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 47 + 38\cdot 59 + 42\cdot 59^{2} + 8\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 29 a + 11 + \left(11 a + 14\right)\cdot 59 + \left(52 a + 21\right)\cdot 59^{2} + \left(38 a + 43\right)\cdot 59^{3} + \left(18 a + 44\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 30 a + 40 + \left(47 a + 55\right)\cdot 59 + \left(6 a + 2\right)\cdot 59^{2} + \left(20 a + 30\right)\cdot 59^{3} + \left(40 a + 24\right)\cdot 59^{4} +O\left(59^{ 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.