Artin representation 35.59e20_10859e20.126.1c1

• Label: 35.59e20_10859e20.126.1c1
• Dimension: 35
• Group: $S_7$
• Conductor: $ 59^{20} \cdot 10859^{20}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$35$
Group:	$S_7$
Conductor:	$135780341114777501574811631919182153810572452468949460314845737730073261614229613421018406498836466281709355583549601= 59^{20} \cdot 10859^{20} $
Artin number field:	Splitting field of $f= x^{7} - 2 x^{6} + 2 x^{5} - 3 x^{3} + 2 x^{2} - 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	126
Parity:	Even
Determinant:	1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{2} + 24 x + 2 $

Roots:

$r_{ 1 }$	$=$	$ 21 a + 10 + \left(25 a + 18\right)\cdot 29 + \left(6 a + 7\right)\cdot 29^{2} + \left(a + 23\right)\cdot 29^{3} + \left(5 a + 7\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 18 + 21\cdot 29 + 6\cdot 29^{2} + 15\cdot 29^{3} + 5\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 14 + 7\cdot 29 + 14\cdot 29^{2} + 8\cdot 29^{3} + 28\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 26 a + 21 + \left(18 a + 9\right)\cdot 29 + \left(27 a + 13\right)\cdot 29^{2} + \left(3 a + 6\right)\cdot 29^{3} + \left(17 a + 15\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 21 + 28\cdot 29 + 11\cdot 29^{2} + 12\cdot 29^{3} + 17\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 8 a + 28 + \left(3 a + 9\right)\cdot 29 + \left(22 a + 16\right)\cdot 29^{2} + \left(27 a + 22\right)\cdot 29^{3} + \left(23 a + 2\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 3 a + 6 + \left(10 a + 20\right)\cdot 29 + \left(a + 16\right)\cdot 29^{2} + \left(25 a + 27\right)\cdot 29^{3} + \left(11 a + 9\right)\cdot 29^{4} +O\left(29^{ 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 value
$1$	$1$	$()$	$35$
$21$	$2$	$(1,2)$	$-5$
$105$	$2$	$(1,2)(3,4)(5,6)$	$-1$
$105$	$2$	$(1,2)(3,4)$	$-1$
$70$	$3$	$(1,2,3)$	$-1$
$280$	$3$	$(1,2,3)(4,5,6)$	$-1$
$210$	$4$	$(1,2,3,4)$	$1$
$630$	$4$	$(1,2,3,4)(5,6)$	$1$
$504$	$5$	$(1,2,3,4,5)$	$0$
$210$	$6$	$(1,2,3)(4,5)(6,7)$	$-1$
$420$	$6$	$(1,2,3)(4,5)$	$1$
$840$	$6$	$(1,2,3,4,5,6)$	$-1$
$720$	$7$	$(1,2,3,4,5,6,7)$	$0$
$504$	$10$	$(1,2,3,4,5)(6,7)$	$0$
$420$	$12$	$(1,2,3,4)(5,6,7)$	$1$

The blue line marks the conjugacy class containing complex conjugation.