LMFDB - Artin representation 1.87.14t1.a.d

Basic invariants

Dimension:	$1$
Group:	$C_{14}$
Conductor:	$87$$\medspace = 3 \cdot 29 $
Artin field:	Galois closure of 14.0.22439994995240462987343.1
Galois orbit size:	$6$
Smallest permutation container:	$C_{14}$
Parity:	odd
Dirichlet character:	$\chi_{87}(5,\cdot)$
Projective image:	$C_1$
Projective field:	Galois closure of $\Q$

Defining polynomial

$f(x)$

$=$

$ x^{14} - x^{13} + 16 x^{12} - 17 x^{11} + 66 x^{10} - 84 x^{9} + 38 x^{8} - 315 x^{7} + 65 x^{6} + \cdots + 1681 $

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{7} + 12x + 42 $ x^7 + 12*x + 42

Roots:

$r_{ 1 }$	$=$	$ a^{6} + 18 a^{5} + 4 a^{4} + 4 a^{3} + 11 a^{2} + 34 a + 25 + \left(22 a^{6} + 45 a^{4} + 13 a^{3} + 28 a^{2} + 12 a + 21\right)\cdot 47 + \left(35 a^{6} + 23 a^{5} + 32 a^{4} + 17 a^{3} + 29 a^{2} + 31 a + 23\right)\cdot 47^{2} + \left(36 a^{6} + 23 a^{5} + 10 a^{4} + 39 a^{3} + 7 a^{2} + 4 a + 36\right)\cdot 47^{3} + \left(2 a^{6} + 29 a^{5} + 9 a^{4} + 22 a^{3} + 4 a^{2} + 21 a + 37\right)\cdot 47^{4} +O(47^{5})$ a^6 + 18a^5 + 4a^4 + 4a^3 + 11a^2 + 34a + 25 + (22a^6 + 45a^4 + 13a^3 + 28a^2 + 12a + 21)47 + (35a^6 + 23a^5 + 32a^4 + 17a^3 + 29a^2 + 31a + 23)47^2 + (36a^6 + 23a^5 + 10a^4 + 39a^3 + 7a^2 + 4a + 36)47^3 + (2a^6 + 29a^5 + 9a^4 + 22a^3 + 4a^2 + 21a + 37)47^4+O(47^5)
$r_{ 2 }$	$=$	$ 2 a^{6} + 44 a^{5} + 17 a^{4} + 41 a^{3} + 39 a^{2} + 27 a + 42 + \left(35 a^{6} + 37 a^{5} + 28 a^{4} + 18 a^{3} + 44 a^{2} + 21 a + 27\right)\cdot 47 + \left(15 a^{6} + 11 a^{5} + 29 a^{4} + 9 a^{3} + 6 a^{2} + 2 a + 8\right)\cdot 47^{2} + \left(23 a^{6} + a^{5} + 2 a^{4} + 23 a^{3} + 43 a^{2} + 15 a + 46\right)\cdot 47^{3} + \left(12 a^{6} + 32 a^{5} + 5 a^{4} + 27 a^{3} + 7 a^{2} + 42 a + 36\right)\cdot 47^{4} +O(47^{5})$ 2a^6 + 44a^5 + 17a^4 + 41a^3 + 39a^2 + 27a + 42 + (35a^6 + 37a^5 + 28a^4 + 18a^3 + 44a^2 + 21a + 27)47 + (15a^6 + 11a^5 + 29a^4 + 9a^3 + 6a^2 + 2a + 8)47^2 + (23a^6 + a^5 + 2a^4 + 23a^3 + 43a^2 + 15a + 46)47^3 + (12a^6 + 32a^5 + 5a^4 + 27a^3 + 7a^2 + 42a + 36)*47^4+O(47^5)
$r_{ 3 }$	$=$	$ 4 a^{6} + 12 a^{5} + 41 a^{4} + 28 a^{3} + 45 a^{2} + 45 a + 29 + \left(21 a^{6} + 31 a^{5} + 43 a^{4} + 23 a^{3} + 6 a^{2} + 28 a + 18\right)\cdot 47 + \left(2 a^{6} + 14 a^{5} + 12 a^{4} + 35 a^{3} + 28 a^{2} + 46 a + 46\right)\cdot 47^{2} + \left(25 a^{6} + 18 a^{5} + 14 a^{4} + 20 a^{3} + 46 a^{2} + 34 a + 36\right)\cdot 47^{3} + \left(2 a^{6} + 45 a^{5} + 7 a^{4} + 30 a^{3} + 19 a^{2} + 19 a + 21\right)\cdot 47^{4} +O(47^{5})$ 4a^6 + 12a^5 + 41a^4 + 28a^3 + 45a^2 + 45a + 29 + (21a^6 + 31a^5 + 43a^4 + 23a^3 + 6a^2 + 28a + 18)47 + (2a^6 + 14a^5 + 12a^4 + 35a^3 + 28a^2 + 46a + 46)47^2 + (25a^6 + 18a^5 + 14a^4 + 20a^3 + 46a^2 + 34a + 36)47^3 + (2a^6 + 45a^5 + 7a^4 + 30a^3 + 19a^2 + 19a + 21)47^4+O(47^5)
$r_{ 4 }$	$=$	$ 4 a^{6} + 33 a^{5} + 16 a^{4} + 11 a^{3} + 21 a^{2} + 24 a + 29 + \left(41 a^{6} + 17 a^{5} + 32 a^{4} + 34 a^{3} + 5 a^{2} + 33 a + 29\right)\cdot 47 + \left(16 a^{6} + 35 a^{5} + 13 a^{4} + 18 a^{3} + 44 a^{2} + 38 a + 33\right)\cdot 47^{2} + \left(10 a^{6} + 12 a^{5} + 29 a^{4} + 32 a^{3} + 6 a^{2} + 7 a + 6\right)\cdot 47^{3} + \left(9 a^{6} + 31 a^{4} + 7 a^{3} + 9 a^{2} + 37 a + 30\right)\cdot 47^{4} +O(47^{5})$ 4a^6 + 33a^5 + 16a^4 + 11a^3 + 21a^2 + 24a + 29 + (41a^6 + 17a^5 + 32a^4 + 34a^3 + 5a^2 + 33a + 29)47 + (16a^6 + 35a^5 + 13a^4 + 18a^3 + 44a^2 + 38a + 33)47^2 + (10a^6 + 12a^5 + 29a^4 + 32a^3 + 6a^2 + 7a + 6)47^3 + (9a^6 + 31a^4 + 7a^3 + 9a^2 + 37a + 30)*47^4+O(47^5)
$r_{ 5 }$	$=$	$ 8 a^{6} + 11 a^{5} + 27 a^{4} + 24 a^{3} + 30 a^{2} + 2 a + 14 + \left(21 a^{6} + a^{5} + 20 a^{4} + 8 a^{3} + 21 a^{2} + 38 a + 33\right)\cdot 47 + \left(46 a^{6} + 34 a^{5} + 7 a^{4} + 20 a^{3} + 30 a^{2} + 19 a + 6\right)\cdot 47^{2} + \left(8 a^{6} + 30 a^{5} + 36 a^{4} + 21 a^{3} + 46 a^{2} + 25 a + 31\right)\cdot 47^{3} + \left(9 a^{6} + 4 a^{5} + 42 a^{4} + 45 a^{3} + 36 a + 31\right)\cdot 47^{4} +O(47^{5})$ 8a^6 + 11a^5 + 27a^4 + 24a^3 + 30a^2 + 2a + 14 + (21a^6 + a^5 + 20a^4 + 8a^3 + 21a^2 + 38a + 33)47 + (46a^6 + 34a^5 + 7a^4 + 20a^3 + 30a^2 + 19a + 6)47^2 + (8a^6 + 30a^5 + 36a^4 + 21a^3 + 46a^2 + 25a + 31)47^3 + (9a^6 + 4a^5 + 42a^4 + 45a^3 + 36a + 31)47^4+O(47^5)
$r_{ 6 }$	$=$	$ 22 a^{6} + 41 a^{5} + 28 a^{4} + 6 a^{3} + 44 a^{2} + 25 a + 17 + \left(44 a^{6} + 3 a^{5} + 16 a^{4} + 17 a^{3} + 13 a^{2} + 12 a + 4\right)\cdot 47 + \left(29 a^{6} + 43 a^{5} + 7 a^{4} + 19 a^{3} + 14 a^{2} + 17 a + 45\right)\cdot 47^{2} + \left(43 a^{6} + 39 a^{5} + 35 a^{4} + 41 a^{3} + 30 a^{2} + 9 a + 24\right)\cdot 47^{3} + \left(36 a^{6} + 45 a^{5} + 43 a^{4} + 26 a^{3} + 43 a^{2} + 21 a + 21\right)\cdot 47^{4} +O(47^{5})$ 22a^6 + 41a^5 + 28a^4 + 6a^3 + 44a^2 + 25a + 17 + (44a^6 + 3a^5 + 16a^4 + 17a^3 + 13a^2 + 12a + 4)47 + (29a^6 + 43a^5 + 7a^4 + 19a^3 + 14a^2 + 17a + 45)47^2 + (43a^6 + 39a^5 + 35a^4 + 41a^3 + 30a^2 + 9a + 24)47^3 + (36a^6 + 45a^5 + 43a^4 + 26a^3 + 43a^2 + 21a + 21)47^4+O(47^5)
$r_{ 7 }$	$=$	$ 25 a^{6} + 12 a^{5} + 21 a^{4} + 31 a^{3} + 45 a^{2} + 9 a + 10 + \left(25 a^{6} + 43 a^{5} + 37 a^{4} + 32 a^{3} + 10 a^{2} + 45 a + 44\right)\cdot 47 + \left(44 a^{6} + 9 a^{5} + 41 a^{4} + 9 a^{3} + 8 a^{2} + 45 a + 42\right)\cdot 47^{2} + \left(29 a^{6} + a^{5} + 39 a^{4} + 21 a^{3} + 23 a^{2} + 14 a + 26\right)\cdot 47^{3} + \left(28 a^{6} + 41 a^{5} + 17 a^{4} + a^{3} + 32 a^{2} + 13 a + 1\right)\cdot 47^{4} +O(47^{5})$ 25a^6 + 12a^5 + 21a^4 + 31a^3 + 45a^2 + 9a + 10 + (25a^6 + 43a^5 + 37a^4 + 32a^3 + 10a^2 + 45a + 44)47 + (44a^6 + 9a^5 + 41a^4 + 9a^3 + 8a^2 + 45a + 42)47^2 + (29a^6 + a^5 + 39a^4 + 21a^3 + 23a^2 + 14a + 26)47^3 + (28a^6 + 41a^5 + 17a^4 + a^3 + 32a^2 + 13a + 1)47^4+O(47^5)
$r_{ 8 }$	$=$	$ 25 a^{6} + 20 a^{5} + 33 a^{4} + 16 a^{3} + 22 a^{2} + 15 a + 10 + \left(5 a^{6} + 13 a^{5} + 13 a^{4} + 38 a^{3} + 14 a^{2} + 38 a + 33\right)\cdot 47 + \left(22 a^{6} + 42 a^{5} + 22 a^{4} + 29 a^{3} + 37 a^{2} + 17 a + 13\right)\cdot 47^{2} + \left(35 a^{6} + 31 a^{5} + 4 a^{4} + 15 a^{3} + 37 a^{2} + 32 a + 23\right)\cdot 47^{3} + \left(31 a^{6} + 33 a^{5} + 15 a^{4} + 2 a^{3} + 34 a^{2} + 5 a + 13\right)\cdot 47^{4} +O(47^{5})$ 25a^6 + 20a^5 + 33a^4 + 16a^3 + 22a^2 + 15a + 10 + (5a^6 + 13a^5 + 13a^4 + 38a^3 + 14a^2 + 38a + 33)47 + (22a^6 + 42a^5 + 22a^4 + 29a^3 + 37a^2 + 17a + 13)47^2 + (35a^6 + 31a^5 + 4a^4 + 15a^3 + 37a^2 + 32a + 23)47^3 + (31a^6 + 33a^5 + 15a^4 + 2a^3 + 34a^2 + 5a + 13)47^4+O(47^5)
$r_{ 9 }$	$=$	$ 25 a^{6} + 38 a^{5} + 15 a^{4} + 29 a^{3} + 6 a^{2} + 23 a + 21 + \left(37 a^{6} + 11 a^{5} + 7 a^{4} + 40 a^{3} + 14 a^{2} + 2 a + 40\right)\cdot 47 + \left(17 a^{6} + 8 a^{5} + 38 a^{4} + 2 a^{3} + 45 a^{2} + 20\right)\cdot 47^{2} + \left(40 a^{6} + 33 a^{5} + 41 a^{4} + a^{3} + 38 a^{2} + 33 a + 38\right)\cdot 47^{3} + \left(17 a^{6} + 18 a^{5} + 21 a^{4} + 34 a^{3} + 25 a^{2} + 15 a + 26\right)\cdot 47^{4} +O(47^{5})$ 25a^6 + 38a^5 + 15a^4 + 29a^3 + 6a^2 + 23a + 21 + (37a^6 + 11a^5 + 7a^4 + 40a^3 + 14a^2 + 2a + 40)47 + (17a^6 + 8a^5 + 38a^4 + 2a^3 + 45a^2 + 20)47^2 + (40a^6 + 33a^5 + 41a^4 + a^3 + 38a^2 + 33a + 38)47^3 + (17a^6 + 18a^5 + 21a^4 + 34a^3 + 25a^2 + 15a + 26)47^4+O(47^5)
$r_{ 10 }$	$=$	$ 26 a^{6} + 3 a^{5} + 39 a^{4} + 30 a^{3} + 46 a^{2} + 39 a + 38 + \left(29 a^{6} + 38 a^{5} + 15 a^{4} + a^{3} + 3 a^{2} + 10 a + 18\right)\cdot 47 + \left(5 a^{6} + 11 a^{5} + 33 a^{4} + 17 a^{3} + 4 a^{2} + 34 a + 43\right)\cdot 47^{2} + \left(26 a^{5} + 31 a^{4} + a^{3} + 39 a^{2} + 44 a + 33\right)\cdot 47^{3} + \left(45 a^{5} + 6 a^{3} + 9 a^{2} + 8 a + 17\right)\cdot 47^{4} +O(47^{5})$ 26a^6 + 3a^5 + 39a^4 + 30a^3 + 46a^2 + 39a + 38 + (29a^6 + 38a^5 + 15a^4 + a^3 + 3a^2 + 10a + 18)47 + (5a^6 + 11a^5 + 33a^4 + 17a^3 + 4a^2 + 34a + 43)47^2 + (26a^5 + 31a^4 + a^3 + 39a^2 + 44a + 33)47^3 + (45a^5 + 6a^3 + 9a^2 + 8a + 17)*47^4+O(47^5)
$r_{ 11 }$	$=$	$ 30 a^{6} + a^{5} + 42 a^{4} + 24 a^{3} + a^{2} + 14 a + 12 + \left(35 a^{6} + 16 a^{5} + 5 a^{4} + 27 a^{3} + 30 a^{2} + 5 a + 14\right)\cdot 47 + \left(12 a^{6} + 41 a^{5} + 45 a^{4} + 24 a^{3} + 41 a^{2} + 31 a + 9\right)\cdot 47^{2} + \left(36 a^{6} + 31 a^{5} + 42 a^{4} + 34 a^{3} + 12 a^{2} + 45 a + 43\right)\cdot 47^{3} + \left(32 a^{6} + 18 a^{5} + 40 a^{4} + 7 a^{3} + 41 a^{2} + 21 a + 45\right)\cdot 47^{4} +O(47^{5})$ 30a^6 + a^5 + 42a^4 + 24a^3 + a^2 + 14a + 12 + (35a^6 + 16a^5 + 5a^4 + 27a^3 + 30a^2 + 5a + 14)47 + (12a^6 + 41a^5 + 45a^4 + 24a^3 + 41a^2 + 31a + 9)47^2 + (36a^6 + 31a^5 + 42a^4 + 34a^3 + 12a^2 + 45a + 43)47^3 + (32a^6 + 18a^5 + 40a^4 + 7a^3 + 41a^2 + 21a + 45)47^4+O(47^5)
$r_{ 12 }$	$=$	$ 33 a^{6} + 2 a^{5} + 9 a^{4} + 10 a^{3} + 5 a^{2} + 34 a + 5 + \left(37 a^{6} + 44 a^{5} + 34 a^{4} + 27 a^{3} + 30 a^{2} + 7 a + 35\right)\cdot 47 + \left(3 a^{6} + 3 a^{5} + 34 a^{4} + 20 a^{3} + 33 a^{2} + 5 a + 26\right)\cdot 47^{2} + \left(27 a^{6} + 5 a^{5} + 39 a^{4} + 35 a^{3} + 22 a^{2} + 31 a + 17\right)\cdot 47^{3} + \left(6 a^{6} + 6 a^{5} + 7 a^{4} + a^{3} + 32 a^{2} + a + 16\right)\cdot 47^{4} +O(47^{5})$ 33a^6 + 2a^5 + 9a^4 + 10a^3 + 5a^2 + 34a + 5 + (37a^6 + 44a^5 + 34a^4 + 27a^3 + 30a^2 + 7a + 35)47 + (3a^6 + 3a^5 + 34a^4 + 20a^3 + 33a^2 + 5a + 26)47^2 + (27a^6 + 5a^5 + 39a^4 + 35a^3 + 22a^2 + 31a + 17)47^3 + (6a^6 + 6a^5 + 7a^4 + a^3 + 32a^2 + a + 16)47^4+O(47^5)
$r_{ 13 }$	$=$	$ 37 a^{6} + 28 a^{5} + 10 a^{4} + 34 a^{3} + a^{2} + 38 a + 37 + \left(10 a^{6} + 35 a^{5} + 2 a^{4} + 20 a^{3} + 22 a^{2} + 26 a + 13\right)\cdot 47 + \left(26 a^{6} + 8 a^{5} + 2 a^{4} + 13 a^{3} + 24 a^{2} + 33 a + 20\right)\cdot 47^{2} + \left(20 a^{6} + 21 a^{5} + 3 a^{4} + 4 a^{3} + 23 a^{2} + 28 a + 2\right)\cdot 47^{3} + \left(34 a^{6} + 17 a^{5} + 34 a^{4} + 33 a^{3} + 40 a^{2} + 23 a + 16\right)\cdot 47^{4} +O(47^{5})$ 37a^6 + 28a^5 + 10a^4 + 34a^3 + a^2 + 38a + 37 + (10a^6 + 35a^5 + 2a^4 + 20a^3 + 22a^2 + 26a + 13)47 + (26a^6 + 8a^5 + 2a^4 + 13a^3 + 24a^2 + 33a + 20)47^2 + (20a^6 + 21a^5 + 3a^4 + 4a^3 + 23a^2 + 28a + 2)47^3 + (34a^6 + 17a^5 + 34a^4 + 33a^3 + 40a^2 + 23a + 16)*47^4+O(47^5)
$r_{ 14 }$	$=$	$ 40 a^{6} + 19 a^{5} + 27 a^{4} + 41 a^{3} + 13 a^{2} + 41 + \left(8 a^{6} + 34 a^{5} + 25 a^{4} + 24 a^{3} + 35 a^{2} + 45 a + 40\right)\cdot 47 + \left(2 a^{6} + 40 a^{5} + 7 a^{4} + 43 a^{3} + 27 a^{2} + 4 a + 34\right)\cdot 47^{2} + \left(38 a^{6} + 4 a^{5} + 44 a^{4} + 36 a^{3} + 43 a^{2} + a + 7\right)\cdot 47^{3} + \left(9 a^{6} + 37 a^{5} + 3 a^{4} + 34 a^{3} + 25 a^{2} + 13 a + 11\right)\cdot 47^{4} +O(47^{5})$ 40a^6 + 19a^5 + 27a^4 + 41a^3 + 13a^2 + 41 + (8a^6 + 34a^5 + 25a^4 + 24a^3 + 35a^2 + 45a + 40)47 + (2a^6 + 40a^5 + 7a^4 + 43a^3 + 27a^2 + 4a + 34)47^2 + (38a^6 + 4a^5 + 44a^4 + 36a^3 + 43a^2 + a + 7)47^3 + (9a^6 + 37a^5 + 3a^4 + 34a^3 + 25a^2 + 13a + 11)47^4+O(47^5)

Generators of the action on the roots $r_1, \ldots, r_{ 14 }$

Cycle notation
$(1,5)(2,13)(3,11)(4,14)(6,7)(8,10)(9,12)$
$(1,2,12,3,8,7,4)(5,13,9,11,10,6,14)$

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 14 }$	Character value
$1$	$1$	$()$	$1$
$1$	$2$	$(1,5)(2,13)(3,11)(4,14)(6,7)(8,10)(9,12)$	$-1$
$1$	$7$	$(1,2,12,3,8,7,4)(5,13,9,11,10,6,14)$	$\zeta_{7}^{4}$
$1$	$7$	$(1,12,8,4,2,3,7)(5,9,10,14,13,11,6)$	$\zeta_{7}$
$1$	$7$	$(1,3,4,12,7,2,8)(5,11,14,9,6,13,10)$	$\zeta_{7}^{5}$
$1$	$7$	$(1,8,2,7,12,4,3)(5,10,13,6,9,14,11)$	$\zeta_{7}^{2}$
$1$	$7$	$(1,7,3,2,4,8,12)(5,6,11,13,14,10,9)$	$-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - \zeta_{7} - 1$
$1$	$7$	$(1,4,7,8,3,12,2)(5,14,6,10,11,9,13)$	$\zeta_{7}^{3}$
$1$	$14$	$(1,13,12,11,8,6,4,5,2,9,3,10,7,14)$	$-\zeta_{7}^{4}$
$1$	$14$	$(1,11,4,9,7,13,8,5,3,14,12,6,2,10)$	$-\zeta_{7}^{5}$
$1$	$14$	$(1,6,3,13,4,10,12,5,7,11,2,14,8,9)$	$\zeta_{7}^{5} + \zeta_{7}^{4} + \zeta_{7}^{3} + \zeta_{7}^{2} + \zeta_{7} + 1$
$1$	$14$	$(1,9,8,14,2,11,7,5,12,10,4,13,3,6)$	$-\zeta_{7}$
$1$	$14$	$(1,10,2,6,12,14,3,5,8,13,7,9,4,11)$	$-\zeta_{7}^{2}$
$1$	$14$	$(1,14,7,10,3,9,2,5,4,6,8,11,12,13)$	$-\zeta_{7}^{3}$

The blue line marks the conjugacy class containing complex conjugation.

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Basic invariants

Defining polynomial

Generators of the action on the roots $r_1, \ldots, r_{ 14 }$

Character values on conjugacy classes

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1.87.14t1.a.d
Dimension	$1$
Group	$C_{14}$
Conductor	$87$
Root number	not computed
Indicator	$0$

$r_{ 1 }$	$=$	\( a^{6} + 18 a^{5} + 4 a^{4} + 4 a^{3} + 11 a^{2} + 34 a + 25 + \left(22 a^{6} + 45 a^{4} + 13 a^{3} + 28 a^{2} + 12 a + 21\right)\cdot 47 + \left(35 a^{6} + 23 a^{5} + 32 a^{4} + 17 a^{3} + 29 a^{2} + 31 a + 23\right)\cdot 47^{2} + \left(36 a^{6} + 23 a^{5} + 10 a^{4} + 39 a^{3} + 7 a^{2} + 4 a + 36\right)\cdot 47^{3} + \left(2 a^{6} + 29 a^{5} + 9 a^{4} + 22 a^{3} + 4 a^{2} + 21 a + 37\right)\cdot 47^{4} +O(47^{5})\) a^6 + 18a^5 + 4a^4 + 4a^3 + 11a^2 + 34a + 25 + (22a^6 + 45a^4 + 13a^3 + 28a^2 + 12a + 21)47 + (35a^6 + 23a^5 + 32a^4 + 17a^3 + 29a^2 + 31a + 23)47^2 + (36a^6 + 23a^5 + 10a^4 + 39a^3 + 7a^2 + 4a + 36)47^3 + (2a^6 + 29a^5 + 9a^4 + 22a^3 + 4a^2 + 21a + 37)47^4+O(47^5)
$r_{ 2 }$	$=$	\( 2 a^{6} + 44 a^{5} + 17 a^{4} + 41 a^{3} + 39 a^{2} + 27 a + 42 + \left(35 a^{6} + 37 a^{5} + 28 a^{4} + 18 a^{3} + 44 a^{2} + 21 a + 27\right)\cdot 47 + \left(15 a^{6} + 11 a^{5} + 29 a^{4} + 9 a^{3} + 6 a^{2} + 2 a + 8\right)\cdot 47^{2} + \left(23 a^{6} + a^{5} + 2 a^{4} + 23 a^{3} + 43 a^{2} + 15 a + 46\right)\cdot 47^{3} + \left(12 a^{6} + 32 a^{5} + 5 a^{4} + 27 a^{3} + 7 a^{2} + 42 a + 36\right)\cdot 47^{4} +O(47^{5})\) 2a^6 + 44a^5 + 17a^4 + 41a^3 + 39a^2 + 27a + 42 + (35a^6 + 37a^5 + 28a^4 + 18a^3 + 44a^2 + 21a + 27)47 + (15a^6 + 11a^5 + 29a^4 + 9a^3 + 6a^2 + 2a + 8)47^2 + (23a^6 + a^5 + 2a^4 + 23a^3 + 43a^2 + 15a + 46)47^3 + (12a^6 + 32a^5 + 5a^4 + 27a^3 + 7a^2 + 42a + 36)*47^4+O(47^5)
$r_{ 3 }$	$=$	\( 4 a^{6} + 12 a^{5} + 41 a^{4} + 28 a^{3} + 45 a^{2} + 45 a + 29 + \left(21 a^{6} + 31 a^{5} + 43 a^{4} + 23 a^{3} + 6 a^{2} + 28 a + 18\right)\cdot 47 + \left(2 a^{6} + 14 a^{5} + 12 a^{4} + 35 a^{3} + 28 a^{2} + 46 a + 46\right)\cdot 47^{2} + \left(25 a^{6} + 18 a^{5} + 14 a^{4} + 20 a^{3} + 46 a^{2} + 34 a + 36\right)\cdot 47^{3} + \left(2 a^{6} + 45 a^{5} + 7 a^{4} + 30 a^{3} + 19 a^{2} + 19 a + 21\right)\cdot 47^{4} +O(47^{5})\) 4a^6 + 12a^5 + 41a^4 + 28a^3 + 45a^2 + 45a + 29 + (21a^6 + 31a^5 + 43a^4 + 23a^3 + 6a^2 + 28a + 18)47 + (2a^6 + 14a^5 + 12a^4 + 35a^3 + 28a^2 + 46a + 46)47^2 + (25a^6 + 18a^5 + 14a^4 + 20a^3 + 46a^2 + 34a + 36)47^3 + (2a^6 + 45a^5 + 7a^4 + 30a^3 + 19a^2 + 19a + 21)47^4+O(47^5)
$r_{ 4 }$	$=$	\( 4 a^{6} + 33 a^{5} + 16 a^{4} + 11 a^{3} + 21 a^{2} + 24 a + 29 + \left(41 a^{6} + 17 a^{5} + 32 a^{4} + 34 a^{3} + 5 a^{2} + 33 a + 29\right)\cdot 47 + \left(16 a^{6} + 35 a^{5} + 13 a^{4} + 18 a^{3} + 44 a^{2} + 38 a + 33\right)\cdot 47^{2} + \left(10 a^{6} + 12 a^{5} + 29 a^{4} + 32 a^{3} + 6 a^{2} + 7 a + 6\right)\cdot 47^{3} + \left(9 a^{6} + 31 a^{4} + 7 a^{3} + 9 a^{2} + 37 a + 30\right)\cdot 47^{4} +O(47^{5})\) 4a^6 + 33a^5 + 16a^4 + 11a^3 + 21a^2 + 24a + 29 + (41a^6 + 17a^5 + 32a^4 + 34a^3 + 5a^2 + 33a + 29)47 + (16a^6 + 35a^5 + 13a^4 + 18a^3 + 44a^2 + 38a + 33)47^2 + (10a^6 + 12a^5 + 29a^4 + 32a^3 + 6a^2 + 7a + 6)47^3 + (9a^6 + 31a^4 + 7a^3 + 9a^2 + 37a + 30)*47^4+O(47^5)
$r_{ 5 }$	$=$	\( 8 a^{6} + 11 a^{5} + 27 a^{4} + 24 a^{3} + 30 a^{2} + 2 a + 14 + \left(21 a^{6} + a^{5} + 20 a^{4} + 8 a^{3} + 21 a^{2} + 38 a + 33\right)\cdot 47 + \left(46 a^{6} + 34 a^{5} + 7 a^{4} + 20 a^{3} + 30 a^{2} + 19 a + 6\right)\cdot 47^{2} + \left(8 a^{6} + 30 a^{5} + 36 a^{4} + 21 a^{3} + 46 a^{2} + 25 a + 31\right)\cdot 47^{3} + \left(9 a^{6} + 4 a^{5} + 42 a^{4} + 45 a^{3} + 36 a + 31\right)\cdot 47^{4} +O(47^{5})\) 8a^6 + 11a^5 + 27a^4 + 24a^3 + 30a^2 + 2a + 14 + (21a^6 + a^5 + 20a^4 + 8a^3 + 21a^2 + 38a + 33)47 + (46a^6 + 34a^5 + 7a^4 + 20a^3 + 30a^2 + 19a + 6)47^2 + (8a^6 + 30a^5 + 36a^4 + 21a^3 + 46a^2 + 25a + 31)47^3 + (9a^6 + 4a^5 + 42a^4 + 45a^3 + 36a + 31)47^4+O(47^5)
$r_{ 6 }$	$=$	\( 22 a^{6} + 41 a^{5} + 28 a^{4} + 6 a^{3} + 44 a^{2} + 25 a + 17 + \left(44 a^{6} + 3 a^{5} + 16 a^{4} + 17 a^{3} + 13 a^{2} + 12 a + 4\right)\cdot 47 + \left(29 a^{6} + 43 a^{5} + 7 a^{4} + 19 a^{3} + 14 a^{2} + 17 a + 45\right)\cdot 47^{2} + \left(43 a^{6} + 39 a^{5} + 35 a^{4} + 41 a^{3} + 30 a^{2} + 9 a + 24\right)\cdot 47^{3} + \left(36 a^{6} + 45 a^{5} + 43 a^{4} + 26 a^{3} + 43 a^{2} + 21 a + 21\right)\cdot 47^{4} +O(47^{5})\) 22a^6 + 41a^5 + 28a^4 + 6a^3 + 44a^2 + 25a + 17 + (44a^6 + 3a^5 + 16a^4 + 17a^3 + 13a^2 + 12a + 4)47 + (29a^6 + 43a^5 + 7a^4 + 19a^3 + 14a^2 + 17a + 45)47^2 + (43a^6 + 39a^5 + 35a^4 + 41a^3 + 30a^2 + 9a + 24)47^3 + (36a^6 + 45a^5 + 43a^4 + 26a^3 + 43a^2 + 21a + 21)47^4+O(47^5)
$r_{ 7 }$	$=$	\( 25 a^{6} + 12 a^{5} + 21 a^{4} + 31 a^{3} + 45 a^{2} + 9 a + 10 + \left(25 a^{6} + 43 a^{5} + 37 a^{4} + 32 a^{3} + 10 a^{2} + 45 a + 44\right)\cdot 47 + \left(44 a^{6} + 9 a^{5} + 41 a^{4} + 9 a^{3} + 8 a^{2} + 45 a + 42\right)\cdot 47^{2} + \left(29 a^{6} + a^{5} + 39 a^{4} + 21 a^{3} + 23 a^{2} + 14 a + 26\right)\cdot 47^{3} + \left(28 a^{6} + 41 a^{5} + 17 a^{4} + a^{3} + 32 a^{2} + 13 a + 1\right)\cdot 47^{4} +O(47^{5})\) 25a^6 + 12a^5 + 21a^4 + 31a^3 + 45a^2 + 9a + 10 + (25a^6 + 43a^5 + 37a^4 + 32a^3 + 10a^2 + 45a + 44)47 + (44a^6 + 9a^5 + 41a^4 + 9a^3 + 8a^2 + 45a + 42)47^2 + (29a^6 + a^5 + 39a^4 + 21a^3 + 23a^2 + 14a + 26)47^3 + (28a^6 + 41a^5 + 17a^4 + a^3 + 32a^2 + 13a + 1)47^4+O(47^5)
$r_{ 8 }$	$=$	\( 25 a^{6} + 20 a^{5} + 33 a^{4} + 16 a^{3} + 22 a^{2} + 15 a + 10 + \left(5 a^{6} + 13 a^{5} + 13 a^{4} + 38 a^{3} + 14 a^{2} + 38 a + 33\right)\cdot 47 + \left(22 a^{6} + 42 a^{5} + 22 a^{4} + 29 a^{3} + 37 a^{2} + 17 a + 13\right)\cdot 47^{2} + \left(35 a^{6} + 31 a^{5} + 4 a^{4} + 15 a^{3} + 37 a^{2} + 32 a + 23\right)\cdot 47^{3} + \left(31 a^{6} + 33 a^{5} + 15 a^{4} + 2 a^{3} + 34 a^{2} + 5 a + 13\right)\cdot 47^{4} +O(47^{5})\) 25a^6 + 20a^5 + 33a^4 + 16a^3 + 22a^2 + 15a + 10 + (5a^6 + 13a^5 + 13a^4 + 38a^3 + 14a^2 + 38a + 33)47 + (22a^6 + 42a^5 + 22a^4 + 29a^3 + 37a^2 + 17a + 13)47^2 + (35a^6 + 31a^5 + 4a^4 + 15a^3 + 37a^2 + 32a + 23)47^3 + (31a^6 + 33a^5 + 15a^4 + 2a^3 + 34a^2 + 5a + 13)47^4+O(47^5)
$r_{ 9 }$	$=$	\( 25 a^{6} + 38 a^{5} + 15 a^{4} + 29 a^{3} + 6 a^{2} + 23 a + 21 + \left(37 a^{6} + 11 a^{5} + 7 a^{4} + 40 a^{3} + 14 a^{2} + 2 a + 40\right)\cdot 47 + \left(17 a^{6} + 8 a^{5} + 38 a^{4} + 2 a^{3} + 45 a^{2} + 20\right)\cdot 47^{2} + \left(40 a^{6} + 33 a^{5} + 41 a^{4} + a^{3} + 38 a^{2} + 33 a + 38\right)\cdot 47^{3} + \left(17 a^{6} + 18 a^{5} + 21 a^{4} + 34 a^{3} + 25 a^{2} + 15 a + 26\right)\cdot 47^{4} +O(47^{5})\) 25a^6 + 38a^5 + 15a^4 + 29a^3 + 6a^2 + 23a + 21 + (37a^6 + 11a^5 + 7a^4 + 40a^3 + 14a^2 + 2a + 40)47 + (17a^6 + 8a^5 + 38a^4 + 2a^3 + 45a^2 + 20)47^2 + (40a^6 + 33a^5 + 41a^4 + a^3 + 38a^2 + 33a + 38)47^3 + (17a^6 + 18a^5 + 21a^4 + 34a^3 + 25a^2 + 15a + 26)47^4+O(47^5)
$r_{ 10 }$	$=$	\( 26 a^{6} + 3 a^{5} + 39 a^{4} + 30 a^{3} + 46 a^{2} + 39 a + 38 + \left(29 a^{6} + 38 a^{5} + 15 a^{4} + a^{3} + 3 a^{2} + 10 a + 18\right)\cdot 47 + \left(5 a^{6} + 11 a^{5} + 33 a^{4} + 17 a^{3} + 4 a^{2} + 34 a + 43\right)\cdot 47^{2} + \left(26 a^{5} + 31 a^{4} + a^{3} + 39 a^{2} + 44 a + 33\right)\cdot 47^{3} + \left(45 a^{5} + 6 a^{3} + 9 a^{2} + 8 a + 17\right)\cdot 47^{4} +O(47^{5})\) 26a^6 + 3a^5 + 39a^4 + 30a^3 + 46a^2 + 39a + 38 + (29a^6 + 38a^5 + 15a^4 + a^3 + 3a^2 + 10a + 18)47 + (5a^6 + 11a^5 + 33a^4 + 17a^3 + 4a^2 + 34a + 43)47^2 + (26a^5 + 31a^4 + a^3 + 39a^2 + 44a + 33)47^3 + (45a^5 + 6a^3 + 9a^2 + 8a + 17)*47^4+O(47^5)
$r_{ 11 }$	$=$	\( 30 a^{6} + a^{5} + 42 a^{4} + 24 a^{3} + a^{2} + 14 a + 12 + \left(35 a^{6} + 16 a^{5} + 5 a^{4} + 27 a^{3} + 30 a^{2} + 5 a + 14\right)\cdot 47 + \left(12 a^{6} + 41 a^{5} + 45 a^{4} + 24 a^{3} + 41 a^{2} + 31 a + 9\right)\cdot 47^{2} + \left(36 a^{6} + 31 a^{5} + 42 a^{4} + 34 a^{3} + 12 a^{2} + 45 a + 43\right)\cdot 47^{3} + \left(32 a^{6} + 18 a^{5} + 40 a^{4} + 7 a^{3} + 41 a^{2} + 21 a + 45\right)\cdot 47^{4} +O(47^{5})\) 30a^6 + a^5 + 42a^4 + 24a^3 + a^2 + 14a + 12 + (35a^6 + 16a^5 + 5a^4 + 27a^3 + 30a^2 + 5a + 14)47 + (12a^6 + 41a^5 + 45a^4 + 24a^3 + 41a^2 + 31a + 9)47^2 + (36a^6 + 31a^5 + 42a^4 + 34a^3 + 12a^2 + 45a + 43)47^3 + (32a^6 + 18a^5 + 40a^4 + 7a^3 + 41a^2 + 21a + 45)47^4+O(47^5)
$r_{ 12 }$	$=$	\( 33 a^{6} + 2 a^{5} + 9 a^{4} + 10 a^{3} + 5 a^{2} + 34 a + 5 + \left(37 a^{6} + 44 a^{5} + 34 a^{4} + 27 a^{3} + 30 a^{2} + 7 a + 35\right)\cdot 47 + \left(3 a^{6} + 3 a^{5} + 34 a^{4} + 20 a^{3} + 33 a^{2} + 5 a + 26\right)\cdot 47^{2} + \left(27 a^{6} + 5 a^{5} + 39 a^{4} + 35 a^{3} + 22 a^{2} + 31 a + 17\right)\cdot 47^{3} + \left(6 a^{6} + 6 a^{5} + 7 a^{4} + a^{3} + 32 a^{2} + a + 16\right)\cdot 47^{4} +O(47^{5})\) 33a^6 + 2a^5 + 9a^4 + 10a^3 + 5a^2 + 34a + 5 + (37a^6 + 44a^5 + 34a^4 + 27a^3 + 30a^2 + 7a + 35)47 + (3a^6 + 3a^5 + 34a^4 + 20a^3 + 33a^2 + 5a + 26)47^2 + (27a^6 + 5a^5 + 39a^4 + 35a^3 + 22a^2 + 31a + 17)47^3 + (6a^6 + 6a^5 + 7a^4 + a^3 + 32a^2 + a + 16)47^4+O(47^5)
$r_{ 13 }$	$=$	\( 37 a^{6} + 28 a^{5} + 10 a^{4} + 34 a^{3} + a^{2} + 38 a + 37 + \left(10 a^{6} + 35 a^{5} + 2 a^{4} + 20 a^{3} + 22 a^{2} + 26 a + 13\right)\cdot 47 + \left(26 a^{6} + 8 a^{5} + 2 a^{4} + 13 a^{3} + 24 a^{2} + 33 a + 20\right)\cdot 47^{2} + \left(20 a^{6} + 21 a^{5} + 3 a^{4} + 4 a^{3} + 23 a^{2} + 28 a + 2\right)\cdot 47^{3} + \left(34 a^{6} + 17 a^{5} + 34 a^{4} + 33 a^{3} + 40 a^{2} + 23 a + 16\right)\cdot 47^{4} +O(47^{5})\) 37a^6 + 28a^5 + 10a^4 + 34a^3 + a^2 + 38a + 37 + (10a^6 + 35a^5 + 2a^4 + 20a^3 + 22a^2 + 26a + 13)47 + (26a^6 + 8a^5 + 2a^4 + 13a^3 + 24a^2 + 33a + 20)47^2 + (20a^6 + 21a^5 + 3a^4 + 4a^3 + 23a^2 + 28a + 2)47^3 + (34a^6 + 17a^5 + 34a^4 + 33a^3 + 40a^2 + 23a + 16)*47^4+O(47^5)
$r_{ 14 }$	$=$	\( 40 a^{6} + 19 a^{5} + 27 a^{4} + 41 a^{3} + 13 a^{2} + 41 + \left(8 a^{6} + 34 a^{5} + 25 a^{4} + 24 a^{3} + 35 a^{2} + 45 a + 40\right)\cdot 47 + \left(2 a^{6} + 40 a^{5} + 7 a^{4} + 43 a^{3} + 27 a^{2} + 4 a + 34\right)\cdot 47^{2} + \left(38 a^{6} + 4 a^{5} + 44 a^{4} + 36 a^{3} + 43 a^{2} + a + 7\right)\cdot 47^{3} + \left(9 a^{6} + 37 a^{5} + 3 a^{4} + 34 a^{3} + 25 a^{2} + 13 a + 11\right)\cdot 47^{4} +O(47^{5})\) 40a^6 + 19a^5 + 27a^4 + 41a^3 + 13a^2 + 41 + (8a^6 + 34a^5 + 25a^4 + 24a^3 + 35a^2 + 45a + 40)47 + (2a^6 + 40a^5 + 7a^4 + 43a^3 + 27a^2 + 4a + 34)47^2 + (38a^6 + 4a^5 + 44a^4 + 36a^3 + 43a^2 + a + 7)47^3 + (9a^6 + 37a^5 + 3a^4 + 34a^3 + 25a^2 + 13a + 11)47^4+O(47^5)