Artin representation 14.7e9_139e9_5483e9.30t565.1c1

• Label: 14.7e9_139e9_5483e9.30t565.1c1
• Dimension: 14
• Group: $S_7$
• Conductor: $ 7^{9} \cdot 139^{9} \cdot 5483^{9}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$14$
Group:	$S_7$
Conductor:	$3500900561442135938096267525046311456673706472523766210421039= 7^{9} \cdot 139^{9} \cdot 5483^{9} $
Artin number field:	Splitting field of $f= x^{7} - 3 x^{5} - 3 x^{4} + x^{3} + 4 x^{2} - 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	30T565
Parity:	Odd
Determinant:	1.7_139_5483.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: $ x^{2} + 60 x + 2 $

Roots:

$r_{ 1 }$	$=$	$ 17 a + 25 + 45\cdot 61 + \left(60 a + 29\right)\cdot 61^{2} + \left(39 a + 47\right)\cdot 61^{3} + \left(30 a + 54\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 12 + 15\cdot 61 + 48\cdot 61^{2} + 55\cdot 61^{3} + 31\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 14 a + 11 + \left(28 a + 54\right)\cdot 61 + \left(57 a + 42\right)\cdot 61^{2} + \left(30 a + 53\right)\cdot 61^{3} + \left(31 a + 20\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 47 a + 25 + \left(32 a + 7\right)\cdot 61 + \left(3 a + 11\right)\cdot 61^{2} + \left(30 a + 27\right)\cdot 61^{3} + \left(29 a + 21\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 29 a + 50 + \left(11 a + 24\right)\cdot 61 + \left(44 a + 25\right)\cdot 61^{2} + \left(29 a + 23\right)\cdot 61^{3} + \left(14 a + 42\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 32 a + 18 + \left(49 a + 7\right)\cdot 61 + \left(16 a + 58\right)\cdot 61^{2} + \left(31 a + 8\right)\cdot 61^{3} + \left(46 a + 27\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 44 a + 42 + \left(60 a + 28\right)\cdot 61 + 28\cdot 61^{2} + \left(21 a + 27\right)\cdot 61^{3} + \left(30 a + 45\right)\cdot 61^{4} +O\left(61^{ 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
$1$	$1$	$()$	$14$
$21$	$2$	$(1,2)$	$-4$
$105$	$2$	$(1,2)(3,4)(5,6)$	$0$
$105$	$2$	$(1,2)(3,4)$	$2$
$70$	$3$	$(1,2,3)$	$-1$
$280$	$3$	$(1,2,3)(4,5,6)$	$2$
$210$	$4$	$(1,2,3,4)$	$2$
$630$	$4$	$(1,2,3,4)(5,6)$	$0$
$504$	$5$	$(1,2,3,4,5)$	$-1$
$210$	$6$	$(1,2,3)(4,5)(6,7)$	$-1$
$420$	$6$	$(1,2,3)(4,5)$	$-1$
$840$	$6$	$(1,2,3,4,5,6)$	$0$
$720$	$7$	$(1,2,3,4,5,6,7)$	$0$
$504$	$10$	$(1,2,3,4,5)(6,7)$	$1$
$420$	$12$	$(1,2,3,4)(5,6,7)$	$-1$

The blue line marks the conjugacy class containing complex conjugation.