LMFDB - Artin representation 2.484.15t4.a.d

Basic invariants

Dimension:	$2$
Group:	$S_3 \times C_5$
Conductor:	$484$$\medspace = 2^{2} \cdot 11^{2} $
Artin stem field:	Galois closure of 15.5.35351257235385344.1
Galois orbit size:	$4$
Smallest permutation container:	$S_3 \times C_5$
Parity:	odd
Determinant:	1.11.10t1.a.d
Projective image:	$S_3$
Projective stem field:	Galois closure of 3.1.44.1

Defining polynomial

$f(x)$

$=$

$ x^{15} - 3 x^{13} - 5 x^{12} + 8 x^{11} + x^{10} + 14 x^{9} - 13 x^{8} - 17 x^{7} + 4 x^{6} + x^{5} + \cdots - 1 $

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 6.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{5} + x + 42 $ x^5 + x + 42

Roots:

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

Generators of the action on the roots $r_1, \ldots, r_{ 15 }$

Cycle notation
$(1,10,9,13,8,3,6,12,15,2)(4,5,7,11,14)$
$(2,7)(3,11)(4,12)(5,13)(10,14)$
$(1,11)(4,9)(5,15)(6,14)(7,8)$

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 15 }$	Character value
$1$	$1$	$()$	$2$
$3$	$2$	$(1,3)(2,8)(6,10)(9,12)(13,15)$	$0$
$2$	$3$	$(1,3,11)(2,7,8)(4,9,12)(5,15,13)(6,10,14)$	$-1$
$1$	$5$	$(1,9,8,6,15)(2,10,13,3,12)(4,7,14,5,11)$	$-2 \zeta_{5}^{3} - 2 \zeta_{5}^{2} - 2 \zeta_{5} - 2$
$1$	$5$	$(1,8,15,9,6)(2,13,12,10,3)(4,14,11,7,5)$	$2 \zeta_{5}^{3}$
$1$	$5$	$(1,6,9,15,8)(2,3,10,12,13)(4,5,7,11,14)$	$2 \zeta_{5}^{2}$
$1$	$5$	$(1,15,6,8,9)(2,12,3,13,10)(4,11,5,14,7)$	$2 \zeta_{5}$
$3$	$10$	$(1,10,9,13,8,3,6,12,15,2)(4,5,7,11,14)$	$0$
$3$	$10$	$(1,13,6,2,9,3,15,10,8,12)(4,11,5,14,7)$	$0$
$3$	$10$	$(1,12,8,10,15,3,9,2,6,13)(4,7,14,5,11)$	$0$
$3$	$10$	$(1,2,15,12,6,3,8,13,9,10)(4,14,11,7,5)$	$0$
$2$	$15$	$(1,14,12,15,7,3,6,4,13,8,11,10,9,5,2)$	$-\zeta_{5}^{2}$
$2$	$15$	$(1,12,7,6,13,11,9,2,14,15,3,4,8,10,5)$	$\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$
$2$	$15$	$(1,7,13,9,14,3,8,5,12,6,11,2,15,4,10)$	$-\zeta_{5}^{3}$
$2$	$15$	$(1,13,14,8,12,11,15,10,7,9,3,5,6,2,4)$	$-\zeta_{5}$

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 $r_1, \ldots, r_{ 15 }$

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	2.484.15t4.a.d
Dimension	$2$
Group	$S_3 \times C_5$
Conductor	$484$
Root number	not computed
Indicator	$0$

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