Artin representation 14.242147e4.21t38.1c1

• Label: 14.242147e4.21t38.1c1
• Dimension: 14
• Group: $S_7$
• Conductor: $ 242147^{4}$
• Root number: 1
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$14$
Group:	$S_7$
Conductor:	$3438083115076197212881= 242147^{4} $
Artin number field:	Splitting field of $f= x^{7} - 2 x^{6} + 2 x^{5} - x^{4} - x^{3} + 2 x^{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 $\Q_{ 89 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 89 }$: $ x^{2} + 82 x + 3 $

Roots:

$r_{ 1 }$	$=$	$ 28 + 20\cdot 89 + 50\cdot 89^{2} + 13\cdot 89^{3} + 64\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 67 a + 27 + \left(5 a + 4\right)\cdot 89 + \left(52 a + 6\right)\cdot 89^{2} + \left(38 a + 84\right)\cdot 89^{3} + \left(32 a + 60\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 82 a + \left(41 a + 25\right)\cdot 89 + \left(23 a + 55\right)\cdot 89^{2} + \left(26 a + 58\right)\cdot 89^{3} + \left(23 a + 22\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 7 a + 40 + \left(47 a + 58\right)\cdot 89 + \left(65 a + 88\right)\cdot 89^{2} + \left(62 a + 40\right)\cdot 89^{3} + \left(65 a + 70\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 22 a + 51 + \left(83 a + 66\right)\cdot 89 + \left(36 a + 8\right)\cdot 89^{2} + \left(50 a + 35\right)\cdot 89^{3} + \left(56 a + 71\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 29 + 13\cdot 89 + 18\cdot 89^{2} + 11\cdot 89^{3} + 83\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 5 + 79\cdot 89 + 39\cdot 89^{2} + 23\cdot 89^{3} + 72\cdot 89^{4} +O\left(89^{ 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.