Artin representation 35.13e15_19259e15.70.1c1

• Label: 35.13e15_19259e15.70.1c1
• Dimension: 35
• Group: $S_7$
• Conductor: $ 13^{15} \cdot 19259^{15}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$35$
Group:	$S_7$
Conductor:	$952042381564610881306939728976613976694679535260934929207355220208319571239980543= 13^{15} \cdot 19259^{15} $
Artin number field:	Splitting field of $f= x^{7} - 2 x^{6} + 4 x^{5} - 4 x^{4} + 3 x^{3} - x^{2} - x + 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	70
Parity:	Odd
Determinant:	1.13_19259.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 101 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 101 }$: $ x^{2} + 97 x + 2 $

Roots:

$r_{ 1 }$	$=$	$ 18 a + 14 + \left(24 a + 17\right)\cdot 101 + \left(82 a + 3\right)\cdot 101^{2} + \left(12 a + 63\right)\cdot 101^{3} + \left(69 a + 59\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 74 a + 31 + \left(54 a + 26\right)\cdot 101 + \left(70 a + 80\right)\cdot 101^{2} + \left(16 a + 100\right)\cdot 101^{3} + 24\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 2 a + 87 + \left(43 a + 68\right)\cdot 101 + \left(17 a + 44\right)\cdot 101^{2} + \left(9 a + 95\right)\cdot 101^{3} + \left(34 a + 91\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 83 a + 86 + \left(76 a + 95\right)\cdot 101 + \left(18 a + 4\right)\cdot 101^{2} + \left(88 a + 32\right)\cdot 101^{3} + \left(31 a + 20\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 69 + 88\cdot 101 + 93\cdot 101^{2} + 101^{3} + 80\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 27 a + 24 + \left(46 a + 70\right)\cdot 101 + \left(30 a + 4\right)\cdot 101^{2} + \left(84 a + 97\right)\cdot 101^{3} + \left(100 a + 8\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 99 a + 95 + \left(57 a + 36\right)\cdot 101 + \left(83 a + 71\right)\cdot 101^{2} + \left(91 a + 13\right)\cdot 101^{3} + \left(66 a + 17\right)\cdot 101^{4} +O\left(101^{ 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.