LMFDB - Artin representation 5.974875729.12t183.a.a

Basic invariants

Dimension:	$5$
Group:	$S_6$
Conductor:	$974875729$$\medspace = 31223^{2} $
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin stem field:	Galois closure of 6.0.31223.1
Galois orbit size:	$1$
Smallest permutation container:	12T183
Parity:	even
Determinant:	1.1.1t1.a.a
Projective image:	$S_6$
Projective stem field:	Galois closure of 6.0.31223.1

$f(x)$

$=$

$ x^{6} - x^{5} + x^{4} - x^{3} + 2x^{2} - x + 1 $

x^6 - x^5 + x^4 - x^3 + 2*x^2 - x + 1

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} + 82x + 3 $ x^2 + 82*x + 3

Roots:

$r_{ 1 }$	$=$	$ 55 a + 45 + \left(70 a + 11\right)\cdot 89 + \left(12 a + 67\right)\cdot 89^{2} + \left(60 a + 88\right)\cdot 89^{3} + \left(67 a + 53\right)\cdot 89^{4} +O(89^{5})$ 55a + 45 + (70a + 11)89 + (12a + 67)89^2 + (60a + 88)89^3 + (67a + 53)*89^4+O(89^5)
$r_{ 2 }$	$=$	$ 34 a + 74 + \left(18 a + 5\right)\cdot 89 + \left(76 a + 86\right)\cdot 89^{2} + \left(28 a + 51\right)\cdot 89^{3} + \left(21 a + 22\right)\cdot 89^{4} +O(89^{5})$ 34a + 74 + (18a + 5)89 + (76a + 86)89^2 + (28a + 51)89^3 + (21a + 22)*89^4+O(89^5)
$r_{ 3 }$	$=$	$ 54 + 37\cdot 89 + 42\cdot 89^{3} + 23\cdot 89^{4} +O(89^{5})$ 54 + 3789 + 4289^3 + 23*89^4+O(89^5)
$r_{ 4 }$	$=$	$ 42 a + 32 + \left(33 a + 14\right)\cdot 89 + \left(57 a + 85\right)\cdot 89^{2} + \left(57 a + 87\right)\cdot 89^{3} + \left(68 a + 72\right)\cdot 89^{4} +O(89^{5})$ 42a + 32 + (33a + 14)89 + (57a + 85)89^2 + (57a + 87)89^3 + (68a + 72)*89^4+O(89^5)
$r_{ 5 }$	$=$	$ 4 + 80\cdot 89 + 19\cdot 89^{2} + 7\cdot 89^{3} + 43\cdot 89^{4} +O(89^{5})$ 4 + 8089 + 1989^2 + 789^3 + 4389^4+O(89^5)
$r_{ 6 }$	$=$	$ 47 a + 59 + \left(55 a + 28\right)\cdot 89 + \left(31 a + 8\right)\cdot 89^{2} + \left(31 a + 78\right)\cdot 89^{3} + \left(20 a + 50\right)\cdot 89^{4} +O(89^{5})$ 47a + 59 + (55a + 28)89 + (31a + 8)89^2 + (31a + 78)89^3 + (20a + 50)*89^4+O(89^5)

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

$r_{ 1 }$	$=$	\( 55 a + 45 + \left(70 a + 11\right)\cdot 89 + \left(12 a + 67\right)\cdot 89^{2} + \left(60 a + 88\right)\cdot 89^{3} + \left(67 a + 53\right)\cdot 89^{4} +O(89^{5})\) 55a + 45 + (70a + 11)89 + (12a + 67)89^2 + (60a + 88)89^3 + (67a + 53)*89^4+O(89^5)
$r_{ 2 }$	$=$	\( 34 a + 74 + \left(18 a + 5\right)\cdot 89 + \left(76 a + 86\right)\cdot 89^{2} + \left(28 a + 51\right)\cdot 89^{3} + \left(21 a + 22\right)\cdot 89^{4} +O(89^{5})\) 34a + 74 + (18a + 5)89 + (76a + 86)89^2 + (28a + 51)89^3 + (21a + 22)*89^4+O(89^5)
$r_{ 3 }$	$=$	\( 54 + 37\cdot 89 + 42\cdot 89^{3} + 23\cdot 89^{4} +O(89^{5})\) 54 + 3789 + 4289^3 + 23*89^4+O(89^5)
$r_{ 4 }$	$=$	\( 42 a + 32 + \left(33 a + 14\right)\cdot 89 + \left(57 a + 85\right)\cdot 89^{2} + \left(57 a + 87\right)\cdot 89^{3} + \left(68 a + 72\right)\cdot 89^{4} +O(89^{5})\) 42a + 32 + (33a + 14)89 + (57a + 85)89^2 + (57a + 87)89^3 + (68a + 72)*89^4+O(89^5)
$r_{ 5 }$	$=$	\( 4 + 80\cdot 89 + 19\cdot 89^{2} + 7\cdot 89^{3} + 43\cdot 89^{4} +O(89^{5})\) 4 + 8089 + 1989^2 + 789^3 + 4389^4+O(89^5)
$r_{ 6 }$	$=$	\( 47 a + 59 + \left(55 a + 28\right)\cdot 89 + \left(31 a + 8\right)\cdot 89^{2} + \left(31 a + 78\right)\cdot 89^{3} + \left(20 a + 50\right)\cdot 89^{4} +O(89^{5})\) 47a + 59 + (55a + 28)89 + (31a + 8)89^2 + (31a + 78)89^3 + (20a + 50)*89^4+O(89^5)