Artin representation 14.3e4_251e4_69073e4.21t38.1c1

• Label: 14.3e4_251e4_69073e4.21t38.1c1
• Dimension: 14
• Group: $S_7$
• Conductor: $ 3^{4} \cdot 251^{4} \cdot 69073^{4}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$14$
Group:	$S_7$
Conductor:	$7318350073165601440328346955521= 3^{4} \cdot 251^{4} \cdot 69073^{4} $
Artin number field:	Splitting field of $f= x^{7} - 3 x^{6} - 3 x^{5} + 13 x^{4} - x^{3} - 13 x^{2} + 4 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 $\Q_{ 239 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 239 }$: $ x^{2} + 237 x + 7 $

Roots:

$r_{ 1 }$	$=$	$ 7 a + 6 + \left(145 a + 28\right)\cdot 239 + \left(209 a + 121\right)\cdot 239^{2} + \left(204 a + 32\right)\cdot 239^{3} + \left(233 a + 77\right)\cdot 239^{4} +O\left(239^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 217 a + 199 + \left(92 a + 213\right)\cdot 239 + \left(144 a + 26\right)\cdot 239^{2} + \left(64 a + 112\right)\cdot 239^{3} + \left(60 a + 149\right)\cdot 239^{4} +O\left(239^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 232 a + 20 + \left(93 a + 72\right)\cdot 239 + \left(29 a + 156\right)\cdot 239^{2} + \left(34 a + 232\right)\cdot 239^{3} + \left(5 a + 100\right)\cdot 239^{4} +O\left(239^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 134 + 93\cdot 239 + 122\cdot 239^{2} + 117\cdot 239^{3} + 200\cdot 239^{4} +O\left(239^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 22 a + 155 + \left(146 a + 182\right)\cdot 239 + \left(94 a + 222\right)\cdot 239^{2} + \left(174 a + 96\right)\cdot 239^{3} + \left(178 a + 205\right)\cdot 239^{4} +O\left(239^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 27 + 235\cdot 239 + 68\cdot 239^{2} + 41\cdot 239^{3} + 195\cdot 239^{4} +O\left(239^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 179 + 130\cdot 239 + 237\cdot 239^{2} + 83\cdot 239^{3} + 27\cdot 239^{4} +O\left(239^{ 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.