Artin representation 14.4297259e4.21t38.1c1

• Label: 14.4297259e4.21t38.1c1
• Dimension: 14
• Group: $S_7$
• Conductor: $ 4297259^{4}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$14$
Group:	$S_7$
Conductor:	$341009218399056880024912561= 4297259^{4} $
Artin number field:	Splitting field of $f= x^{7} - 3 x^{5} - 3 x^{4} + 4 x^{3} + 5 x^{2} - 2 x - 1 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	21T38
Parity:	Even
Determinant:	1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 157 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 157 }$: $ x^{2} + 152 x + 5 $

Roots:

$r_{ 1 }$	$=$	$ 26 a + 149 + \left(50 a + 42\right)\cdot 157 + \left(60 a + 47\right)\cdot 157^{2} + \left(30 a + 51\right)\cdot 157^{3} + \left(114 a + 99\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 22 + 78\cdot 157 + 104\cdot 157^{2} + 117\cdot 157^{3} + 85\cdot 157^{4} +O\left(157^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 63 a + 138 + \left(48 a + 98\right)\cdot 157 + \left(35 a + 102\right)\cdot 157^{2} + \left(94 a + 154\right)\cdot 157^{3} + \left(109 a + 4\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 106 + 68\cdot 157 + 131\cdot 157^{2} + 145\cdot 157^{3} + 10\cdot 157^{4} +O\left(157^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 109 + 107\cdot 157 + 26\cdot 157^{2} + 53\cdot 157^{3} + 113\cdot 157^{4} +O\left(157^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 131 a + 122 + \left(106 a + 110\right)\cdot 157 + \left(96 a + 141\right)\cdot 157^{2} + \left(126 a + 142\right)\cdot 157^{3} + \left(42 a + 11\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 94 a + 139 + \left(108 a + 120\right)\cdot 157 + \left(121 a + 73\right)\cdot 157^{2} + \left(62 a + 119\right)\cdot 157^{3} + \left(47 a + 144\right)\cdot 157^{4} +O\left(157^{ 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
$1$	$1$	$()$	$14$
$21$	$2$	$(1,2)$	$6$
$105$	$2$	$(1,2)(3,4)(5,6)$	$2$
$105$	$2$	$(1,2)(3,4)$	$2$
$70$	$3$	$(1,2,3)$	$2$
$280$	$3$	$(1,2,3)(4,5,6)$	$-1$
$210$	$4$	$(1,2,3,4)$	$0$
$630$	$4$	$(1,2,3,4)(5,6)$	$0$
$504$	$5$	$(1,2,3,4,5)$	$-1$
$210$	$6$	$(1,2,3)(4,5)(6,7)$	$2$
$420$	$6$	$(1,2,3)(4,5)$	$0$
$840$	$6$	$(1,2,3,4,5,6)$	$-1$
$720$	$7$	$(1,2,3,4,5,6,7)$	$0$
$504$	$10$	$(1,2,3,4,5)(6,7)$	$1$
$420$	$12$	$(1,2,3,4)(5,6,7)$	$0$

The blue line marks the conjugacy class containing complex conjugation.