LMFDB - Artin representation 14.3438083115076197212881.21t38.a.a

Basic invariants

Dimension:	$14$
Group:	$S_7$
Conductor:	$343\!\cdots\!881$$\medspace = 242147^{4} $
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin stem field:	Galois closure of 7.1.242147.1
Galois orbit size:	$1$
Smallest permutation container:	21T38
Parity:	even
Determinant:	1.1.1t1.a.a
Projective image:	$S_7$
Projective stem field:	Galois closure of 7.1.242147.1

$f(x)$

$=$

$ x^{7} - 2x^{6} + 2x^{5} - x^{4} - x^{3} + 2x^{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:

$r_{ 1 }$	$=$	$ 28 + 20\cdot 89 + 50\cdot 89^{2} + 13\cdot 89^{3} + 64\cdot 89^{4} +O(89^{5})$ 28 + 2089 + 5089^2 + 1389^3 + 6489^4+O(89^5)
$r_{ 2 }$	$=$	$ 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(89^{5})$ 67a + 27 + (5a + 4)89 + (52a + 6)89^2 + (38a + 84)89^3 + (32a + 60)*89^4+O(89^5)
$r_{ 3 }$	$=$	$ 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(89^{5})$ 82a + (41a + 25)89 + (23a + 55)89^2 + (26a + 58)89^3 + (23a + 22)*89^4+O(89^5)
$r_{ 4 }$	$=$	$ 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(89^{5})$ 7a + 40 + (47a + 58)89 + (65a + 88)89^2 + (62a + 40)89^3 + (65a + 70)*89^4+O(89^5)
$r_{ 5 }$	$=$	$ 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(89^{5})$ 22a + 51 + (83a + 66)89 + (36a + 8)89^2 + (50a + 35)89^3 + (56a + 71)*89^4+O(89^5)
$r_{ 6 }$	$=$	$ 29 + 13\cdot 89 + 18\cdot 89^{2} + 11\cdot 89^{3} + 83\cdot 89^{4} +O(89^{5})$ 29 + 1389 + 1889^2 + 1189^3 + 8389^4+O(89^5)
$r_{ 7 }$	$=$	$ 5 + 79\cdot 89 + 39\cdot 89^{2} + 23\cdot 89^{3} + 72\cdot 89^{4} +O(89^{5})$ 5 + 7989 + 3989^2 + 2389^3 + 7289^4+O(89^5)

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

$r_{ 1 }$	$=$	\( 28 + 20\cdot 89 + 50\cdot 89^{2} + 13\cdot 89^{3} + 64\cdot 89^{4} +O(89^{5})\) 28 + 2089 + 5089^2 + 1389^3 + 6489^4+O(89^5)
$r_{ 2 }$	$=$	\( 67 a + 27 + \left(5 a + 4\right)\cdot 89 + \left(52 a + 6\right)\cdot 89^{2} + \left(38 a + 84\right)\cdot 89^{3} + \left(32 a + 60\right)\cdot 89^{4} +O(89^{5})\) 67a + 27 + (5a + 4)89 + (52a + 6)89^2 + (38a + 84)89^3 + (32a + 60)*89^4+O(89^5)
$r_{ 3 }$	$=$	\( 82 a + \left(41 a + 25\right)\cdot 89 + \left(23 a + 55\right)\cdot 89^{2} + \left(26 a + 58\right)\cdot 89^{3} + \left(23 a + 22\right)\cdot 89^{4} +O(89^{5})\) 82a + (41a + 25)89 + (23a + 55)89^2 + (26a + 58)89^3 + (23a + 22)*89^4+O(89^5)
$r_{ 4 }$	$=$	\( 7 a + 40 + \left(47 a + 58\right)\cdot 89 + \left(65 a + 88\right)\cdot 89^{2} + \left(62 a + 40\right)\cdot 89^{3} + \left(65 a + 70\right)\cdot 89^{4} +O(89^{5})\) 7a + 40 + (47a + 58)89 + (65a + 88)89^2 + (62a + 40)89^3 + (65a + 70)*89^4+O(89^5)
$r_{ 5 }$	$=$	\( 22 a + 51 + \left(83 a + 66\right)\cdot 89 + \left(36 a + 8\right)\cdot 89^{2} + \left(50 a + 35\right)\cdot 89^{3} + \left(56 a + 71\right)\cdot 89^{4} +O(89^{5})\) 22a + 51 + (83a + 66)89 + (36a + 8)89^2 + (50a + 35)89^3 + (56a + 71)*89^4+O(89^5)
$r_{ 6 }$	$=$	\( 29 + 13\cdot 89 + 18\cdot 89^{2} + 11\cdot 89^{3} + 83\cdot 89^{4} +O(89^{5})\) 29 + 1389 + 1889^2 + 1189^3 + 8389^4+O(89^5)
$r_{ 7 }$	$=$	\( 5 + 79\cdot 89 + 39\cdot 89^{2} + 23\cdot 89^{3} + 72\cdot 89^{4} +O(89^{5})\) 5 + 7989 + 3989^2 + 2389^3 + 7289^4+O(89^5)