LMFDB - Artin representation 2.2303.15t2.a.a

Basic invariants

Dimension:	$2$
Group:	$D_{15}$
Conductor:	$2303$$\medspace = 7^{2} \cdot 47 $
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin stem field:	Galois closure of 15.1.143108492101942920287.1
Galois orbit size:	$4$
Smallest permutation container:	$D_{15}$
Parity:	odd
Determinant:	1.47.2t1.a.a
Projective image:	$D_{15}$
Projective stem field:	Galois closure of 15.1.143108492101942920287.1

Defining polynomial

$f(x)$

$=$

$ x^{15} - x^{14} + 3 x^{13} - 21 x^{12} + 23 x^{11} + 8 x^{10} + 39 x^{9} - 113 x^{8} - 64 x^{7} + \cdots + 13 $

The roots of $f$ are computed in an extension of $\Q_{ 79 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 79 }$: $ x^{5} + 5x + 76 $ x^5 + 5*x + 76

Roots:

$r_{ 1 }$	$=$	$ 14 a^{3} + 32 a^{2} + 60 a + 37 + \left(42 a^{4} + 70 a^{3} + 54 a^{2} + 32 a + 55\right)\cdot 79 + \left(9 a^{4} + 29 a^{3} + 64 a^{2} + 45 a + 62\right)\cdot 79^{2} + \left(46 a^{4} + 45 a^{3} + 50 a^{2} + 20 a + 39\right)\cdot 79^{3} + \left(43 a^{4} + 63 a^{3} + 70 a^{2} + 58 a + 19\right)\cdot 79^{4} +O(79^{5})$ 14a^3 + 32a^2 + 60a + 37 + (42a^4 + 70a^3 + 54a^2 + 32a + 55)79 + (9a^4 + 29a^3 + 64a^2 + 45a + 62)79^2 + (46a^4 + 45a^3 + 50a^2 + 20a + 39)79^3 + (43a^4 + 63a^3 + 70a^2 + 58a + 19)*79^4+O(79^5)
$r_{ 2 }$	$=$	$ 5 a^{4} + 61 a^{3} + 65 a^{2} + 13 a + 57 + \left(43 a^{4} + 34 a^{3} + 15 a^{2} + 2 a + 59\right)\cdot 79 + \left(15 a^{4} + 5 a^{3} + 40 a^{2} + 36 a + 7\right)\cdot 79^{2} + \left(55 a^{4} + 18 a^{3} + 15 a^{2} + 38 a + 76\right)\cdot 79^{3} + \left(74 a^{4} + 19 a^{3} + 68 a^{2} + 37 a + 64\right)\cdot 79^{4} +O(79^{5})$ 5a^4 + 61a^3 + 65a^2 + 13a + 57 + (43a^4 + 34a^3 + 15a^2 + 2a + 59)79 + (15a^4 + 5a^3 + 40a^2 + 36a + 7)79^2 + (55a^4 + 18a^3 + 15a^2 + 38a + 76)79^3 + (74a^4 + 19a^3 + 68a^2 + 37a + 64)79^4+O(79^5)
$r_{ 3 }$	$=$	$ 11 a^{4} + 40 a^{3} + 42 a^{2} + 43 a + 11 + \left(25 a^{4} + 51 a^{3} + 77 a^{2} + 74 a + 25\right)\cdot 79 + \left(69 a^{4} + 46 a^{3} + 26 a^{2} + 21 a + 22\right)\cdot 79^{2} + \left(42 a^{4} + 4 a^{3} + 78 a^{2} + 44 a + 25\right)\cdot 79^{3} + \left(68 a^{4} + 76 a^{3} + 36 a^{2} + 48 a + 70\right)\cdot 79^{4} +O(79^{5})$ 11a^4 + 40a^3 + 42a^2 + 43a + 11 + (25a^4 + 51a^3 + 77a^2 + 74a + 25)79 + (69a^4 + 46a^3 + 26a^2 + 21a + 22)79^2 + (42a^4 + 4a^3 + 78a^2 + 44a + 25)79^3 + (68a^4 + 76a^3 + 36a^2 + 48a + 70)79^4+O(79^5)
$r_{ 4 }$	$=$	$ 16 a^{4} + 18 a^{3} + 5 a^{2} + 55 a + 22 + \left(55 a^{4} + 50 a^{3} + 42 a^{2} + 20 a + 29\right)\cdot 79 + \left(51 a^{4} + 38 a^{3} + 71 a^{2} + 40 a + 73\right)\cdot 79^{2} + \left(a^{4} + 44 a^{3} + 5 a^{2} + 27 a + 19\right)\cdot 79^{3} + \left(19 a^{4} + 51 a^{3} + 71 a^{2} + 60 a\right)\cdot 79^{4} +O(79^{5})$ 16a^4 + 18a^3 + 5a^2 + 55a + 22 + (55a^4 + 50a^3 + 42a^2 + 20a + 29)79 + (51a^4 + 38a^3 + 71a^2 + 40a + 73)79^2 + (a^4 + 44a^3 + 5a^2 + 27a + 19)79^3 + (19a^4 + 51a^3 + 71a^2 + 60a)*79^4+O(79^5)
$r_{ 5 }$	$=$	$ 18 a^{4} + 9 a^{3} + 70 a^{2} + 39 + \left(69 a^{4} + 73 a^{3} + 33 a^{2} + 41 a + 43\right)\cdot 79 + \left(7 a^{4} + 55 a^{3} + 16 a^{2} + 8 a + 13\right)\cdot 79^{2} + \left(31 a^{4} + 64 a^{3} + 50 a^{2} + 22 a + 57\right)\cdot 79^{3} + \left(78 a^{4} + 34 a^{3} + 10 a^{2} + 34 a + 30\right)\cdot 79^{4} +O(79^{5})$ 18a^4 + 9a^3 + 70a^2 + 39 + (69a^4 + 73a^3 + 33a^2 + 41a + 43)79 + (7a^4 + 55a^3 + 16a^2 + 8a + 13)79^2 + (31a^4 + 64a^3 + 50a^2 + 22a + 57)79^3 + (78a^4 + 34a^3 + 10a^2 + 34a + 30)*79^4+O(79^5)
$r_{ 6 }$	$=$	$ 24 a^{4} + 23 a^{3} + 30 a^{2} + 34 a + 29 + \left(26 a^{4} + 42 a^{3} + 76 a^{2} + 66 a + 40\right)\cdot 79 + \left(39 a^{4} + 66 a^{3} + 31 a^{2} + 5 a + 8\right)\cdot 79^{2} + \left(76 a^{4} + 41 a^{3} + 54 a^{2} + 40 a + 28\right)\cdot 79^{3} + \left(39 a^{4} + 49 a^{3} + 23 a^{2} + 65 a + 60\right)\cdot 79^{4} +O(79^{5})$ 24a^4 + 23a^3 + 30a^2 + 34a + 29 + (26a^4 + 42a^3 + 76a^2 + 66a + 40)79 + (39a^4 + 66a^3 + 31a^2 + 5a + 8)79^2 + (76a^4 + 41a^3 + 54a^2 + 40a + 28)79^3 + (39a^4 + 49a^3 + 23a^2 + 65a + 60)79^4+O(79^5)
$r_{ 7 }$	$=$	$ 24 a^{4} + 35 a^{3} + 44 a^{2} + 18 a + 54 + \left(3 a^{4} + 34 a^{3} + 71 a^{2} + 32 a + 58\right)\cdot 79 + \left(77 a^{4} + 59 a^{3} + 5 a^{2} + 9 a + 16\right)\cdot 79^{2} + \left(5 a^{4} + 13 a^{3} + 67 a^{2} + 45 a + 37\right)\cdot 79^{3} + \left(44 a^{4} + 11 a^{3} + 60 a^{2} + 52 a + 21\right)\cdot 79^{4} +O(79^{5})$ 24a^4 + 35a^3 + 44a^2 + 18a + 54 + (3a^4 + 34a^3 + 71a^2 + 32a + 58)79 + (77a^4 + 59a^3 + 5a^2 + 9a + 16)79^2 + (5a^4 + 13a^3 + 67a^2 + 45a + 37)79^3 + (44a^4 + 11a^3 + 60a^2 + 52a + 21)79^4+O(79^5)
$r_{ 8 }$	$=$	$ 29 a^{4} + 11 a^{3} + 76 a^{2} + 6 a + 4 + \left(15 a^{4} + 54 a^{3} + 38 a^{2} + 53 a + 65\right)\cdot 79 + \left(26 a^{4} + 57 a^{3} + 27 a^{2} + 75 a + 7\right)\cdot 79^{2} + \left(51 a^{4} + 33 a^{3} + 9 a^{2} + 73 a + 59\right)\cdot 79^{3} + \left(8 a^{4} + 26 a^{3} + 41 a + 67\right)\cdot 79^{4} +O(79^{5})$ 29a^4 + 11a^3 + 76a^2 + 6a + 4 + (15a^4 + 54a^3 + 38a^2 + 53a + 65)79 + (26a^4 + 57a^3 + 27a^2 + 75a + 7)79^2 + (51a^4 + 33a^3 + 9a^2 + 73a + 59)79^3 + (8a^4 + 26a^3 + 41a + 67)*79^4+O(79^5)
$r_{ 9 }$	$=$	$ 30 a^{4} + 36 a^{3} + a^{2} + 4 a + 53 + \left(35 a^{4} + 47 a^{3} + 3 a^{2} + 76\right)\cdot 79 + \left(68 a^{4} + 33 a^{3} + 27 a^{2} + 4 a + 45\right)\cdot 79^{2} + \left(77 a^{4} + 15 a^{3} + a^{2} + 56 a + 33\right)\cdot 79^{3} + \left(57 a^{4} + 32 a^{3} + 8 a^{2} + 37 a + 53\right)\cdot 79^{4} +O(79^{5})$ 30a^4 + 36a^3 + a^2 + 4a + 53 + (35a^4 + 47a^3 + 3a^2 + 76)79 + (68a^4 + 33a^3 + 27a^2 + 4a + 45)79^2 + (77a^4 + 15a^3 + a^2 + 56a + 33)79^3 + (57a^4 + 32a^3 + 8a^2 + 37a + 53)*79^4+O(79^5)
$r_{ 10 }$	$=$	$ 34 a^{4} + 30 a^{3} + 12 a^{2} + 12 a + 15 + \left(14 a^{4} + 47 a^{3} + 53 a^{2} + 70 a + 24\right)\cdot 79 + \left(4 a^{4} + 24 a^{3} + 54 a^{2} + 26 a + 41\right)\cdot 79^{2} + \left(49 a^{4} + 36 a^{3} + 18 a^{2} + 26 a + 51\right)\cdot 79^{3} + \left(55 a^{4} + 12 a^{3} + 45 a^{2} + 28 a + 67\right)\cdot 79^{4} +O(79^{5})$ 34a^4 + 30a^3 + 12a^2 + 12a + 15 + (14a^4 + 47a^3 + 53a^2 + 70a + 24)79 + (4a^4 + 24a^3 + 54a^2 + 26a + 41)79^2 + (49a^4 + 36a^3 + 18a^2 + 26a + 51)79^3 + (55a^4 + 12a^3 + 45a^2 + 28a + 67)79^4+O(79^5)
$r_{ 11 }$	$=$	$ 38 a^{4} + 17 a^{3} + 62 a^{2} + 22 a + 40 + \left(56 a^{4} + 76 a^{3} + 14 a^{2} + a + 71\right)\cdot 79 + \left(78 a^{4} + a^{3} + 28 a^{2} + 69 a + 59\right)\cdot 79^{2} + \left(72 a^{4} + 74 a^{2} + 48 a + 66\right)\cdot 79^{3} + \left(25 a^{4} + 12 a^{3} + 28 a^{2} + 68 a + 57\right)\cdot 79^{4} +O(79^{5})$ 38a^4 + 17a^3 + 62a^2 + 22a + 40 + (56a^4 + 76a^3 + 14a^2 + a + 71)79 + (78a^4 + a^3 + 28a^2 + 69a + 59)79^2 + (72a^4 + 74a^2 + 48a + 66)79^3 + (25a^4 + 12a^3 + 28a^2 + 68a + 57)*79^4+O(79^5)
$r_{ 12 }$	$=$	$ 39 a^{4} + 67 a^{3} + 73 a^{2} + 12 a + 10 + \left(21 a^{4} + 31 a^{3} + 19 a^{2} + 76 a + 21\right)\cdot 79 + \left(55 a^{4} + 70 a^{3} + 37 a^{2} + 76 a + 72\right)\cdot 79^{2} + \left(40 a^{4} + 69 a^{3} + 42 a^{2} + 36 a + 42\right)\cdot 79^{3} + \left(64 a^{4} + 42 a^{3} + 69 a^{2} + 24 a\right)\cdot 79^{4} +O(79^{5})$ 39a^4 + 67a^3 + 73a^2 + 12a + 10 + (21a^4 + 31a^3 + 19a^2 + 76a + 21)79 + (55a^4 + 70a^3 + 37a^2 + 76a + 72)79^2 + (40a^4 + 69a^3 + 42a^2 + 36a + 42)79^3 + (64a^4 + 42a^3 + 69a^2 + 24a)79^4+O(79^5)
$r_{ 13 }$	$=$	$ 62 a^{4} + 2 a^{3} + 66 a^{2} + 8 a + 57 + \left(70 a^{4} + 61 a^{3} + 71 a^{2} + 67 a + 49\right)\cdot 79 + \left(54 a^{4} + 74 a^{3} + 58 a^{2} + 61 a + 43\right)\cdot 79^{2} + \left(38 a^{4} + 54 a^{3} + 24 a^{2} + 47 a + 8\right)\cdot 79^{3} + \left(55 a^{4} + 8 a^{3} + 2 a^{2} + 43 a + 18\right)\cdot 79^{4} +O(79^{5})$ 62a^4 + 2a^3 + 66a^2 + 8a + 57 + (70a^4 + 61a^3 + 71a^2 + 67a + 49)79 + (54a^4 + 74a^3 + 58a^2 + 61a + 43)79^2 + (38a^4 + 54a^3 + 24a^2 + 47a + 8)79^3 + (55a^4 + 8a^3 + 2a^2 + 43a + 18)79^4+O(79^5)
$r_{ 14 }$	$=$	$ 71 a^{4} + 7 a^{3} + 31 a^{2} + 23 a + 59 + \left(77 a^{4} + 3 a^{3} + 33 a^{2} + 68 a + 9\right)\cdot 79 + \left(61 a^{4} + 6 a^{3} + 46 a^{2} + 2 a + 20\right)\cdot 79^{2} + \left(66 a^{4} + 51 a^{3} + 49 a^{2} + 56 a + 68\right)\cdot 79^{3} + \left(a^{4} + 43 a^{3} + 7 a^{2} + 13 a + 65\right)\cdot 79^{4} +O(79^{5})$ 71a^4 + 7a^3 + 31a^2 + 23a + 59 + (77a^4 + 3a^3 + 33a^2 + 68a + 9)79 + (61a^4 + 6a^3 + 46a^2 + 2a + 20)79^2 + (66a^4 + 51a^3 + 49a^2 + 56a + 68)79^3 + (a^4 + 43a^3 + 7a^2 + 13a + 65)*79^4+O(79^5)
$r_{ 15 }$	$=$	$ 73 a^{4} + 25 a^{3} + 23 a^{2} + 6 a + 67 + \left(75 a^{4} + 33 a^{3} + 25 a^{2} + 26 a + 1\right)\cdot 79 + \left(11 a^{4} + 60 a^{3} + 15 a^{2} + 68 a + 57\right)\cdot 79^{2} + \left(54 a^{4} + 58 a^{3} + 10 a^{2} + 47 a + 17\right)\cdot 79^{3} + \left(72 a^{4} + 68 a^{3} + 49 a^{2} + 16 a + 33\right)\cdot 79^{4} +O(79^{5})$ 73a^4 + 25a^3 + 23a^2 + 6a + 67 + (75a^4 + 33a^3 + 25a^2 + 26a + 1)79 + (11a^4 + 60a^3 + 15a^2 + 68a + 57)79^2 + (54a^4 + 58a^3 + 10a^2 + 47a + 17)79^3 + (72a^4 + 68a^3 + 49a^2 + 16a + 33)79^4+O(79^5)

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

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

Character values on conjugacy classes

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

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	2.2303.15t2.a.a
Dimension	$2$
Group	$D_{15}$
Conductor	$2303$
Root number	$1$
Indicator	$1$

$r_{ 1 }$	$=$	\( 14 a^{3} + 32 a^{2} + 60 a + 37 + \left(42 a^{4} + 70 a^{3} + 54 a^{2} + 32 a + 55\right)\cdot 79 + \left(9 a^{4} + 29 a^{3} + 64 a^{2} + 45 a + 62\right)\cdot 79^{2} + \left(46 a^{4} + 45 a^{3} + 50 a^{2} + 20 a + 39\right)\cdot 79^{3} + \left(43 a^{4} + 63 a^{3} + 70 a^{2} + 58 a + 19\right)\cdot 79^{4} +O(79^{5})\) 14a^3 + 32a^2 + 60a + 37 + (42a^4 + 70a^3 + 54a^2 + 32a + 55)79 + (9a^4 + 29a^3 + 64a^2 + 45a + 62)79^2 + (46a^4 + 45a^3 + 50a^2 + 20a + 39)79^3 + (43a^4 + 63a^3 + 70a^2 + 58a + 19)*79^4+O(79^5)
$r_{ 2 }$	$=$	\( 5 a^{4} + 61 a^{3} + 65 a^{2} + 13 a + 57 + \left(43 a^{4} + 34 a^{3} + 15 a^{2} + 2 a + 59\right)\cdot 79 + \left(15 a^{4} + 5 a^{3} + 40 a^{2} + 36 a + 7\right)\cdot 79^{2} + \left(55 a^{4} + 18 a^{3} + 15 a^{2} + 38 a + 76\right)\cdot 79^{3} + \left(74 a^{4} + 19 a^{3} + 68 a^{2} + 37 a + 64\right)\cdot 79^{4} +O(79^{5})\) 5a^4 + 61a^3 + 65a^2 + 13a + 57 + (43a^4 + 34a^3 + 15a^2 + 2a + 59)79 + (15a^4 + 5a^3 + 40a^2 + 36a + 7)79^2 + (55a^4 + 18a^3 + 15a^2 + 38a + 76)79^3 + (74a^4 + 19a^3 + 68a^2 + 37a + 64)79^4+O(79^5)
$r_{ 3 }$	$=$	\( 11 a^{4} + 40 a^{3} + 42 a^{2} + 43 a + 11 + \left(25 a^{4} + 51 a^{3} + 77 a^{2} + 74 a + 25\right)\cdot 79 + \left(69 a^{4} + 46 a^{3} + 26 a^{2} + 21 a + 22\right)\cdot 79^{2} + \left(42 a^{4} + 4 a^{3} + 78 a^{2} + 44 a + 25\right)\cdot 79^{3} + \left(68 a^{4} + 76 a^{3} + 36 a^{2} + 48 a + 70\right)\cdot 79^{4} +O(79^{5})\) 11a^4 + 40a^3 + 42a^2 + 43a + 11 + (25a^4 + 51a^3 + 77a^2 + 74a + 25)79 + (69a^4 + 46a^3 + 26a^2 + 21a + 22)79^2 + (42a^4 + 4a^3 + 78a^2 + 44a + 25)79^3 + (68a^4 + 76a^3 + 36a^2 + 48a + 70)79^4+O(79^5)
$r_{ 4 }$	$=$	\( 16 a^{4} + 18 a^{3} + 5 a^{2} + 55 a + 22 + \left(55 a^{4} + 50 a^{3} + 42 a^{2} + 20 a + 29\right)\cdot 79 + \left(51 a^{4} + 38 a^{3} + 71 a^{2} + 40 a + 73\right)\cdot 79^{2} + \left(a^{4} + 44 a^{3} + 5 a^{2} + 27 a + 19\right)\cdot 79^{3} + \left(19 a^{4} + 51 a^{3} + 71 a^{2} + 60 a\right)\cdot 79^{4} +O(79^{5})\) 16a^4 + 18a^3 + 5a^2 + 55a + 22 + (55a^4 + 50a^3 + 42a^2 + 20a + 29)79 + (51a^4 + 38a^3 + 71a^2 + 40a + 73)79^2 + (a^4 + 44a^3 + 5a^2 + 27a + 19)79^3 + (19a^4 + 51a^3 + 71a^2 + 60a)*79^4+O(79^5)
$r_{ 5 }$	$=$	\( 18 a^{4} + 9 a^{3} + 70 a^{2} + 39 + \left(69 a^{4} + 73 a^{3} + 33 a^{2} + 41 a + 43\right)\cdot 79 + \left(7 a^{4} + 55 a^{3} + 16 a^{2} + 8 a + 13\right)\cdot 79^{2} + \left(31 a^{4} + 64 a^{3} + 50 a^{2} + 22 a + 57\right)\cdot 79^{3} + \left(78 a^{4} + 34 a^{3} + 10 a^{2} + 34 a + 30\right)\cdot 79^{4} +O(79^{5})\) 18a^4 + 9a^3 + 70a^2 + 39 + (69a^4 + 73a^3 + 33a^2 + 41a + 43)79 + (7a^4 + 55a^3 + 16a^2 + 8a + 13)79^2 + (31a^4 + 64a^3 + 50a^2 + 22a + 57)79^3 + (78a^4 + 34a^3 + 10a^2 + 34a + 30)*79^4+O(79^5)
$r_{ 6 }$	$=$	\( 24 a^{4} + 23 a^{3} + 30 a^{2} + 34 a + 29 + \left(26 a^{4} + 42 a^{3} + 76 a^{2} + 66 a + 40\right)\cdot 79 + \left(39 a^{4} + 66 a^{3} + 31 a^{2} + 5 a + 8\right)\cdot 79^{2} + \left(76 a^{4} + 41 a^{3} + 54 a^{2} + 40 a + 28\right)\cdot 79^{3} + \left(39 a^{4} + 49 a^{3} + 23 a^{2} + 65 a + 60\right)\cdot 79^{4} +O(79^{5})\) 24a^4 + 23a^3 + 30a^2 + 34a + 29 + (26a^4 + 42a^3 + 76a^2 + 66a + 40)79 + (39a^4 + 66a^3 + 31a^2 + 5a + 8)79^2 + (76a^4 + 41a^3 + 54a^2 + 40a + 28)79^3 + (39a^4 + 49a^3 + 23a^2 + 65a + 60)79^4+O(79^5)
$r_{ 7 }$	$=$	\( 24 a^{4} + 35 a^{3} + 44 a^{2} + 18 a + 54 + \left(3 a^{4} + 34 a^{3} + 71 a^{2} + 32 a + 58\right)\cdot 79 + \left(77 a^{4} + 59 a^{3} + 5 a^{2} + 9 a + 16\right)\cdot 79^{2} + \left(5 a^{4} + 13 a^{3} + 67 a^{2} + 45 a + 37\right)\cdot 79^{3} + \left(44 a^{4} + 11 a^{3} + 60 a^{2} + 52 a + 21\right)\cdot 79^{4} +O(79^{5})\) 24a^4 + 35a^3 + 44a^2 + 18a + 54 + (3a^4 + 34a^3 + 71a^2 + 32a + 58)79 + (77a^4 + 59a^3 + 5a^2 + 9a + 16)79^2 + (5a^4 + 13a^3 + 67a^2 + 45a + 37)79^3 + (44a^4 + 11a^3 + 60a^2 + 52a + 21)79^4+O(79^5)
$r_{ 8 }$	$=$	\( 29 a^{4} + 11 a^{3} + 76 a^{2} + 6 a + 4 + \left(15 a^{4} + 54 a^{3} + 38 a^{2} + 53 a + 65\right)\cdot 79 + \left(26 a^{4} + 57 a^{3} + 27 a^{2} + 75 a + 7\right)\cdot 79^{2} + \left(51 a^{4} + 33 a^{3} + 9 a^{2} + 73 a + 59\right)\cdot 79^{3} + \left(8 a^{4} + 26 a^{3} + 41 a + 67\right)\cdot 79^{4} +O(79^{5})\) 29a^4 + 11a^3 + 76a^2 + 6a + 4 + (15a^4 + 54a^3 + 38a^2 + 53a + 65)79 + (26a^4 + 57a^3 + 27a^2 + 75a + 7)79^2 + (51a^4 + 33a^3 + 9a^2 + 73a + 59)79^3 + (8a^4 + 26a^3 + 41a + 67)*79^4+O(79^5)
$r_{ 9 }$	$=$	\( 30 a^{4} + 36 a^{3} + a^{2} + 4 a + 53 + \left(35 a^{4} + 47 a^{3} + 3 a^{2} + 76\right)\cdot 79 + \left(68 a^{4} + 33 a^{3} + 27 a^{2} + 4 a + 45\right)\cdot 79^{2} + \left(77 a^{4} + 15 a^{3} + a^{2} + 56 a + 33\right)\cdot 79^{3} + \left(57 a^{4} + 32 a^{3} + 8 a^{2} + 37 a + 53\right)\cdot 79^{4} +O(79^{5})\) 30a^4 + 36a^3 + a^2 + 4a + 53 + (35a^4 + 47a^3 + 3a^2 + 76)79 + (68a^4 + 33a^3 + 27a^2 + 4a + 45)79^2 + (77a^4 + 15a^3 + a^2 + 56a + 33)79^3 + (57a^4 + 32a^3 + 8a^2 + 37a + 53)*79^4+O(79^5)
$r_{ 10 }$	$=$	\( 34 a^{4} + 30 a^{3} + 12 a^{2} + 12 a + 15 + \left(14 a^{4} + 47 a^{3} + 53 a^{2} + 70 a + 24\right)\cdot 79 + \left(4 a^{4} + 24 a^{3} + 54 a^{2} + 26 a + 41\right)\cdot 79^{2} + \left(49 a^{4} + 36 a^{3} + 18 a^{2} + 26 a + 51\right)\cdot 79^{3} + \left(55 a^{4} + 12 a^{3} + 45 a^{2} + 28 a + 67\right)\cdot 79^{4} +O(79^{5})\) 34a^4 + 30a^3 + 12a^2 + 12a + 15 + (14a^4 + 47a^3 + 53a^2 + 70a + 24)79 + (4a^4 + 24a^3 + 54a^2 + 26a + 41)79^2 + (49a^4 + 36a^3 + 18a^2 + 26a + 51)79^3 + (55a^4 + 12a^3 + 45a^2 + 28a + 67)79^4+O(79^5)
$r_{ 11 }$	$=$	\( 38 a^{4} + 17 a^{3} + 62 a^{2} + 22 a + 40 + \left(56 a^{4} + 76 a^{3} + 14 a^{2} + a + 71\right)\cdot 79 + \left(78 a^{4} + a^{3} + 28 a^{2} + 69 a + 59\right)\cdot 79^{2} + \left(72 a^{4} + 74 a^{2} + 48 a + 66\right)\cdot 79^{3} + \left(25 a^{4} + 12 a^{3} + 28 a^{2} + 68 a + 57\right)\cdot 79^{4} +O(79^{5})\) 38a^4 + 17a^3 + 62a^2 + 22a + 40 + (56a^4 + 76a^3 + 14a^2 + a + 71)79 + (78a^4 + a^3 + 28a^2 + 69a + 59)79^2 + (72a^4 + 74a^2 + 48a + 66)79^3 + (25a^4 + 12a^3 + 28a^2 + 68a + 57)*79^4+O(79^5)
$r_{ 12 }$	$=$	\( 39 a^{4} + 67 a^{3} + 73 a^{2} + 12 a + 10 + \left(21 a^{4} + 31 a^{3} + 19 a^{2} + 76 a + 21\right)\cdot 79 + \left(55 a^{4} + 70 a^{3} + 37 a^{2} + 76 a + 72\right)\cdot 79^{2} + \left(40 a^{4} + 69 a^{3} + 42 a^{2} + 36 a + 42\right)\cdot 79^{3} + \left(64 a^{4} + 42 a^{3} + 69 a^{2} + 24 a\right)\cdot 79^{4} +O(79^{5})\) 39a^4 + 67a^3 + 73a^2 + 12a + 10 + (21a^4 + 31a^3 + 19a^2 + 76a + 21)79 + (55a^4 + 70a^3 + 37a^2 + 76a + 72)79^2 + (40a^4 + 69a^3 + 42a^2 + 36a + 42)79^3 + (64a^4 + 42a^3 + 69a^2 + 24a)79^4+O(79^5)
$r_{ 13 }$	$=$	\( 62 a^{4} + 2 a^{3} + 66 a^{2} + 8 a + 57 + \left(70 a^{4} + 61 a^{3} + 71 a^{2} + 67 a + 49\right)\cdot 79 + \left(54 a^{4} + 74 a^{3} + 58 a^{2} + 61 a + 43\right)\cdot 79^{2} + \left(38 a^{4} + 54 a^{3} + 24 a^{2} + 47 a + 8\right)\cdot 79^{3} + \left(55 a^{4} + 8 a^{3} + 2 a^{2} + 43 a + 18\right)\cdot 79^{4} +O(79^{5})\) 62a^4 + 2a^3 + 66a^2 + 8a + 57 + (70a^4 + 61a^3 + 71a^2 + 67a + 49)79 + (54a^4 + 74a^3 + 58a^2 + 61a + 43)79^2 + (38a^4 + 54a^3 + 24a^2 + 47a + 8)79^3 + (55a^4 + 8a^3 + 2a^2 + 43a + 18)79^4+O(79^5)
$r_{ 14 }$	$=$	\( 71 a^{4} + 7 a^{3} + 31 a^{2} + 23 a + 59 + \left(77 a^{4} + 3 a^{3} + 33 a^{2} + 68 a + 9\right)\cdot 79 + \left(61 a^{4} + 6 a^{3} + 46 a^{2} + 2 a + 20\right)\cdot 79^{2} + \left(66 a^{4} + 51 a^{3} + 49 a^{2} + 56 a + 68\right)\cdot 79^{3} + \left(a^{4} + 43 a^{3} + 7 a^{2} + 13 a + 65\right)\cdot 79^{4} +O(79^{5})\) 71a^4 + 7a^3 + 31a^2 + 23a + 59 + (77a^4 + 3a^3 + 33a^2 + 68a + 9)79 + (61a^4 + 6a^3 + 46a^2 + 2a + 20)79^2 + (66a^4 + 51a^3 + 49a^2 + 56a + 68)79^3 + (a^4 + 43a^3 + 7a^2 + 13a + 65)*79^4+O(79^5)
$r_{ 15 }$	$=$	\( 73 a^{4} + 25 a^{3} + 23 a^{2} + 6 a + 67 + \left(75 a^{4} + 33 a^{3} + 25 a^{2} + 26 a + 1\right)\cdot 79 + \left(11 a^{4} + 60 a^{3} + 15 a^{2} + 68 a + 57\right)\cdot 79^{2} + \left(54 a^{4} + 58 a^{3} + 10 a^{2} + 47 a + 17\right)\cdot 79^{3} + \left(72 a^{4} + 68 a^{3} + 49 a^{2} + 16 a + 33\right)\cdot 79^{4} +O(79^{5})\) 73a^4 + 25a^3 + 23a^2 + 6a + 67 + (75a^4 + 33a^3 + 25a^2 + 26a + 1)79 + (11a^4 + 60a^3 + 15a^2 + 68a + 57)79^2 + (54a^4 + 58a^3 + 10a^2 + 47a + 17)79^3 + (72a^4 + 68a^3 + 49a^2 + 16a + 33)79^4+O(79^5)