LMFDB - Artin representation 2.3375.15t2.a.d

Basic invariants

Dimension:	$2$
Group:	$D_{15}$
Conductor:	$3375$$\medspace = 3^{3} \cdot 5^{3} $
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin stem field:	Galois closure of 15.1.2463153133392333984375.1
Galois orbit size:	$4$
Smallest permutation container:	$D_{15}$
Parity:	odd
Determinant:	1.15.2t1.a.a
Projective image:	$D_{15}$
Projective stem field:	Galois closure of 15.1.2463153133392333984375.1

Defining polynomial

$f(x)$

$=$

$ x^{15} - 10 x^{12} + 15 x^{11} - 24 x^{10} + 95 x^{9} - 90 x^{8} + 90 x^{7} - 125 x^{6} + 27 x^{5} + \cdots + 33 $

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:

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

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

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

Character values on conjugacy classes

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

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	2.3375.15t2.a.d
Dimension	$2$
Group	$D_{15}$
Conductor	$3375$
Root number	$1$
Indicator	$1$

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