Artin representation 35.37e15_20357e15.70.1c1

• Label: 35.37e15_20357e15.70.1c1
• Dimension: 35
• Group: $S_7$
• Conductor: $ 37^{15} \cdot 20357^{15}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$35$
Group:	$S_7$
Conductor:	$14247298078168197049948699718275179059761885322977892476348509064795601796743501762272649= 37^{15} \cdot 20357^{15} $
Artin number field:	Splitting field of $f= x^{7} - x^{6} - x^{5} + 4 x^{4} - 3 x^{3} - x^{2} + 3 x - 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	70
Parity:	Even
Determinant:	1.37_20357.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$: $ x^{2} + 69 x + 7 $

Roots:

$r_{ 1 }$	$=$	$ 28 a + 45 + \left(27 a + 49\right)\cdot 71 + \left(6 a + 17\right)\cdot 71^{2} + \left(2 a + 56\right)\cdot 71^{3} + \left(6 a + 68\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 55 a + 21 + \left(14 a + 38\right)\cdot 71 + \left(2 a + 53\right)\cdot 71^{2} + \left(20 a + 22\right)\cdot 71^{3} + \left(68 a + 57\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 47 a + 56 + \left(59 a + 22\right)\cdot 71 + 8 a\cdot 71^{2} + \left(19 a + 34\right)\cdot 71^{3} + \left(14 a + 27\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 16 a + 60 + \left(56 a + 12\right)\cdot 71 + \left(68 a + 43\right)\cdot 71^{2} + \left(50 a + 60\right)\cdot 71^{3} + \left(2 a + 31\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 43 a + 30 + \left(43 a + 5\right)\cdot 71 + \left(64 a + 3\right)\cdot 71^{2} + \left(68 a + 54\right)\cdot 71^{3} + \left(64 a + 7\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 24 a + 8 + \left(11 a + 24\right)\cdot 71 + \left(62 a + 29\right)\cdot 71^{2} + \left(51 a + 63\right)\cdot 71^{3} + \left(56 a + 36\right)\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 65 + 59\cdot 71 + 65\cdot 71^{2} + 63\cdot 71^{3} + 53\cdot 71^{4} +O\left(71^{ 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.