LMFDB - Artin representation 2.2159.15t2.a.d

Basic invariants

Dimension:	$2$
Group:	$D_{15}$
Conductor:	$2159$$\medspace = 17 \cdot 127 $
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin stem field:	Galois closure of 15.1.218659573334046061397519.1
Galois orbit size:	$4$
Smallest permutation container:	$D_{15}$
Parity:	odd
Determinant:	1.2159.2t1.a.a
Projective image:	$D_{15}$
Projective stem field:	Galois closure of 15.1.218659573334046061397519.1

Defining polynomial

$f(x)$

$=$

$ x^{15} - 15 x^{12} + 14 x^{11} + 101 x^{10} + 110 x^{9} - 304 x^{8} - 603 x^{7} - 60 x^{6} + 921 x^{5} + \cdots + 1 $

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^{5} + x + 42 $ x^5 + x + 42

Roots:

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

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

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

Character values on conjugacy classes

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

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	2.2159.15t2.a.d
Dimension	$2$
Group	$D_{15}$
Conductor	$2159$
Root number	$1$
Indicator	$1$

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