LMFDB - Artin representation 2.1452.15t4.a.c

Basic invariants

Dimension:	$2$
Group:	$S_3 \times C_5$
Conductor:	$1452$$\medspace = 2^{2} \cdot 3 \cdot 11^{2} $
Artin stem field:	Galois closure of 15.5.94493910590185024512.1
Galois orbit size:	$4$
Smallest permutation container:	$S_3 \times C_5$
Parity:	odd
Determinant:	1.33.10t1.a.b
Projective image:	$S_3$
Projective stem field:	Galois closure of 3.1.1452.1

Defining polynomial

$f(x)$

$=$

$ x^{15} - 5 x^{14} + 8 x^{13} + 8 x^{12} - 52 x^{11} + 77 x^{10} - 22 x^{9} + 11 x^{8} - 121 x^{7} + \cdots + 43 $

The roots of $f$ are computed in an extension of $\Q_{ 139 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 139 }$: $ x^{5} + 10x + 137 $ x^5 + 10*x + 137

Roots:

$r_{ 1 }$	$=$	$ 18 a^{4} + 13 a^{3} + 54 a^{2} + 42 a + 121 + \left(3 a^{4} + 29 a^{3} + 90 a^{2} + 71 a + 73\right)\cdot 139 + \left(24 a^{4} + 82 a^{3} + 56 a^{2} + 112 a + 55\right)\cdot 139^{2} + \left(120 a^{4} + 44 a^{3} + 26 a^{2} + 13 a + 31\right)\cdot 139^{3} + \left(43 a^{4} + 35 a^{3} + 120 a^{2} + 73 a + 17\right)\cdot 139^{4} +O(139^{5})$ 18a^4 + 13a^3 + 54a^2 + 42a + 121 + (3a^4 + 29a^3 + 90a^2 + 71a + 73)139 + (24a^4 + 82a^3 + 56a^2 + 112a + 55)139^2 + (120a^4 + 44a^3 + 26a^2 + 13a + 31)139^3 + (43a^4 + 35a^3 + 120a^2 + 73a + 17)139^4+O(139^5)
$r_{ 2 }$	$=$	$ 27 a^{4} + 68 a^{3} + 75 a^{2} + 54 a + 23 + \left(121 a^{4} + 27 a^{3} + 34 a^{2} + 74 a + 73\right)\cdot 139 + \left(49 a^{4} + 88 a^{3} + 32 a^{2} + 134 a + 98\right)\cdot 139^{2} + \left(43 a^{4} + 120 a^{3} + 55 a^{2} + 18 a + 121\right)\cdot 139^{3} + \left(22 a^{4} + 91 a^{3} + 136 a^{2} + 23 a + 42\right)\cdot 139^{4} +O(139^{5})$ 27a^4 + 68a^3 + 75a^2 + 54a + 23 + (121a^4 + 27a^3 + 34a^2 + 74a + 73)139 + (49a^4 + 88a^3 + 32a^2 + 134a + 98)139^2 + (43a^4 + 120a^3 + 55a^2 + 18a + 121)139^3 + (22a^4 + 91a^3 + 136a^2 + 23a + 42)139^4+O(139^5)
$r_{ 3 }$	$=$	$ 33 a^{4} + 13 a^{3} + 114 a^{2} + 138 a + 102 + \left(53 a^{4} + 25 a^{3} + 72 a^{2} + 31 a + 57\right)\cdot 139 + \left(67 a^{4} + 10 a^{3} + 44 a^{2} + 44 a + 124\right)\cdot 139^{2} + \left(44 a^{4} + 125 a^{3} + 127 a^{2} + 83 a + 120\right)\cdot 139^{3} + \left(128 a^{4} + 102 a^{3} + 3 a^{2} + 62 a + 136\right)\cdot 139^{4} +O(139^{5})$ 33a^4 + 13a^3 + 114a^2 + 138a + 102 + (53a^4 + 25a^3 + 72a^2 + 31a + 57)139 + (67a^4 + 10a^3 + 44a^2 + 44a + 124)139^2 + (44a^4 + 125a^3 + 127a^2 + 83a + 120)139^3 + (128a^4 + 102a^3 + 3a^2 + 62a + 136)139^4+O(139^5)
$r_{ 4 }$	$=$	$ 33 a^{4} + 125 a^{3} + 72 a^{2} + 106 a + 64 + \left(68 a^{4} + 129 a^{3} + 28 a^{2} + 121 a + 3\right)\cdot 139 + \left(17 a^{4} + 28 a^{3} + 110 a^{2} + 11 a + 21\right)\cdot 139^{2} + \left(83 a^{4} + 78 a^{3} + 9 a^{2} + 11 a + 13\right)\cdot 139^{3} + \left(70 a^{4} + 134 a^{3} + 40 a^{2} + 102 a + 61\right)\cdot 139^{4} +O(139^{5})$ 33a^4 + 125a^3 + 72a^2 + 106a + 64 + (68a^4 + 129a^3 + 28a^2 + 121a + 3)139 + (17a^4 + 28a^3 + 110a^2 + 11a + 21)139^2 + (83a^4 + 78a^3 + 9a^2 + 11a + 13)139^3 + (70a^4 + 134a^3 + 40a^2 + 102a + 61)139^4+O(139^5)
$r_{ 5 }$	$=$	$ 45 a^{4} + 134 a^{3} + 59 a^{2} + 93 a + 28 + \left(66 a^{4} + 78 a^{3} + 92 a^{2} + 109 a + 51\right)\cdot 139 + \left(77 a^{4} + 120 a^{3} + 39 a^{2} + 39 a + 41\right)\cdot 139^{2} + \left(137 a^{4} + 37 a^{3} + 119 a^{2} + 48 a + 41\right)\cdot 139^{3} + \left(102 a^{4} + 83 a^{3} + 134 a^{2} + 132 a + 132\right)\cdot 139^{4} +O(139^{5})$ 45a^4 + 134a^3 + 59a^2 + 93a + 28 + (66a^4 + 78a^3 + 92a^2 + 109a + 51)139 + (77a^4 + 120a^3 + 39a^2 + 39a + 41)139^2 + (137a^4 + 37a^3 + 119a^2 + 48a + 41)139^3 + (102a^4 + 83a^3 + 134a^2 + 132a + 132)139^4+O(139^5)
$r_{ 6 }$	$=$	$ 79 a^{4} + 20 a^{3} + 27 a^{2} + 87 a + 15 + \left(119 a^{4} + 97 a^{3} + 102 a^{2} + 60 a + 136\right)\cdot 139 + \left(41 a^{4} + 25 a^{3} + 68 a^{2} + 132 a + 76\right)\cdot 139^{2} + \left(102 a^{4} + 92 a^{3} + 49 a^{2} + 14 a + 27\right)\cdot 139^{3} + \left(11 a^{4} + 51 a^{3} + 84 a^{2} + 59 a + 7\right)\cdot 139^{4} +O(139^{5})$ 79a^4 + 20a^3 + 27a^2 + 87a + 15 + (119a^4 + 97a^3 + 102a^2 + 60a + 136)139 + (41a^4 + 25a^3 + 68a^2 + 132a + 76)139^2 + (102a^4 + 92a^3 + 49a^2 + 14a + 27)139^3 + (11a^4 + 51a^3 + 84a^2 + 59a + 7)139^4+O(139^5)
$r_{ 7 }$	$=$	$ 85 a^{4} + 113 a^{3} + 103 a^{2} + 115 a + 63 + \left(118 a^{4} + 112 a^{3} + 46 a^{2} + 96 a + 128\right)\cdot 139 + \left(125 a^{4} + 90 a^{3} + 80 a^{2} + 54 a + 53\right)\cdot 139^{2} + \left(81 a^{4} + 14 a^{3} + 14 a^{2} + 72 a + 3\right)\cdot 139^{3} + \left(44 a^{4} + 110 a^{3} + 67 a^{2} + 136 a + 131\right)\cdot 139^{4} +O(139^{5})$ 85a^4 + 113a^3 + 103a^2 + 115a + 63 + (118a^4 + 112a^3 + 46a^2 + 96a + 128)139 + (125a^4 + 90a^3 + 80a^2 + 54a + 53)139^2 + (81a^4 + 14a^3 + 14a^2 + 72a + 3)139^3 + (44a^4 + 110a^3 + 67a^2 + 136a + 131)139^4+O(139^5)
$r_{ 8 }$	$=$	$ 91 a^{4} + 23 a^{3} + 135 a^{2} + 119 a + 118 + \left(24 a^{4} + 68 a^{3} + 91 a^{2} + 110 a + 134\right)\cdot 139 + \left(103 a^{4} + 111 a^{3} + 138 a^{2} + 102 a + 107\right)\cdot 139^{2} + \left(28 a^{4} + 20 a^{3} + 65 a^{2} + 92 a + 4\right)\cdot 139^{3} + \left(29 a^{4} + 42 a^{3} + 31 a^{2} + 50 a + 98\right)\cdot 139^{4} +O(139^{5})$ 91a^4 + 23a^3 + 135a^2 + 119a + 118 + (24a^4 + 68a^3 + 91a^2 + 110a + 134)139 + (103a^4 + 111a^3 + 138a^2 + 102a + 107)139^2 + (28a^4 + 20a^3 + 65a^2 + 92a + 4)139^3 + (29a^4 + 42a^3 + 31a^2 + 50a + 98)139^4+O(139^5)
$r_{ 9 }$	$=$	$ 100 a^{4} + 134 a^{3} + 120 a + 44 + \left(47 a^{4} + 10 a^{3} + 84 a^{2} + 74 a + 117\right)\cdot 139 + \left(59 a^{4} + 114 a^{3} + 19 a^{2} + 134 a + 77\right)\cdot 139^{2} + \left(48 a^{4} + 108 a^{3} + 17 a^{2} + 94 a + 13\right)\cdot 139^{3} + \left(87 a^{4} + 95 a^{3} + 6 a^{2} + 38 a + 56\right)\cdot 139^{4} +O(139^{5})$ 100a^4 + 134a^3 + 120a + 44 + (47a^4 + 10a^3 + 84a^2 + 74a + 117)139 + (59a^4 + 114a^3 + 19a^2 + 134a + 77)139^2 + (48a^4 + 108a^3 + 17a^2 + 94a + 13)139^3 + (87a^4 + 95a^3 + 6a^2 + 38a + 56)*139^4+O(139^5)
$r_{ 10 }$	$=$	$ 111 a^{4} + 26 a^{3} + 100 a^{2} + 35 a + 31 + \left(5 a^{4} + 62 a^{3} + 39 a^{2} + 132 a + 95\right)\cdot 139 + \left(118 a^{4} + 74 a^{3} + 54 a^{2} + 38 a + 112\right)\cdot 139^{2} + \left(38 a^{4} + 108 a^{3} + 54 a^{2} + 88 a + 75\right)\cdot 139^{3} + \left(36 a^{4} + 81 a^{3} + 86 a^{2} + 59 a + 95\right)\cdot 139^{4} +O(139^{5})$ 111a^4 + 26a^3 + 100a^2 + 35a + 31 + (5a^4 + 62a^3 + 39a^2 + 132a + 95)139 + (118a^4 + 74a^3 + 54a^2 + 38a + 112)139^2 + (38a^4 + 108a^3 + 54a^2 + 88a + 75)139^3 + (36a^4 + 81a^3 + 86a^2 + 59a + 95)139^4+O(139^5)
$r_{ 11 }$	$=$	$ 117 a^{4} + 134 a^{3} + 65 a^{2} + 115 a + 79 + \left(57 a^{4} + 125 a^{3} + 135 a^{2} + 112 a + 94\right)\cdot 139 + \left(66 a^{4} + 128 a^{3} + 107 a^{2} + 60 a + 116\right)\cdot 139^{2} + \left(130 a^{4} + 131 a^{3} + 138 a^{2} + 67 a + 113\right)\cdot 139^{3} + \left(61 a^{4} + 78 a^{3} + 126 a^{2} + 78 a + 22\right)\cdot 139^{4} +O(139^{5})$ 117a^4 + 134a^3 + 65a^2 + 115a + 79 + (57a^4 + 125a^3 + 135a^2 + 112a + 94)139 + (66a^4 + 128a^3 + 107a^2 + 60a + 116)139^2 + (130a^4 + 131a^3 + 138a^2 + 67a + 113)139^3 + (61a^4 + 78a^3 + 126a^2 + 78a + 22)139^4+O(139^5)
$r_{ 12 }$	$=$	$ 118 a^{4} + 62 a^{3} + 64 a^{2} + 112 a + 56 + \left(39 a^{4} + 42 a^{3} + 18 a^{2} + 74 a + 117\right)\cdot 139 + \left(26 a^{4} + 83 a^{3} + 133 a^{2} + 75 a + 48\right)\cdot 139^{2} + \left(28 a^{4} + 72 a^{3} + 53 a^{2} + 28 a\right)\cdot 139^{3} + \left(103 a^{4} + 97 a^{3} + 134 a^{2} + 103 a + 134\right)\cdot 139^{4} +O(139^{5})$ 118a^4 + 62a^3 + 64a^2 + 112a + 56 + (39a^4 + 42a^3 + 18a^2 + 74a + 117)139 + (26a^4 + 83a^3 + 133a^2 + 75a + 48)139^2 + (28a^4 + 72a^3 + 53a^2 + 28a)139^3 + (103a^4 + 97a^3 + 134a^2 + 103a + 134)139^4+O(139^5)
$r_{ 13 }$	$=$	$ 120 a^{4} + 25 a^{3} + 76 a^{2} + 128 a + 65 + \left(62 a^{4} + 66 a^{3} + 16 a^{2} + 62 a + 99\right)\cdot 139 + \left(33 a^{4} + 18 a^{3} + 138 a^{2} + 83 a + 9\right)\cdot 139^{2} + \left(101 a^{4} + 123 a^{3} + 47 a^{2} + 84 a + 19\right)\cdot 139^{3} + \left(63 a^{4} + 24 a^{3} + 80 a^{2} + 80 a + 6\right)\cdot 139^{4} +O(139^{5})$ 120a^4 + 25a^3 + 76a^2 + 128a + 65 + (62a^4 + 66a^3 + 16a^2 + 62a + 99)139 + (33a^4 + 18a^3 + 138a^2 + 83a + 9)139^2 + (101a^4 + 123a^3 + 47a^2 + 84a + 19)139^3 + (63a^4 + 24a^3 + 80a^2 + 80a + 6)139^4+O(139^5)
$r_{ 14 }$	$=$	$ 136 a^{4} + 130 a^{3} + 84 a^{2} + 39 a + 61 + \left(25 a^{4} + 60 a^{3} + 40 a^{2} + 47 a + 6\right)\cdot 139 + \left(21 a^{4} + 13 a^{3} + 73 a^{2} + 64 a + 8\right)\cdot 139^{2} + \left(40 a^{4} + 26 a^{3} + 122 a^{2} + 89 a + 96\right)\cdot 139^{3} + \left(20 a^{4} + 102 a^{3} + 118 a^{2} + 107 a + 26\right)\cdot 139^{4} +O(139^{5})$ 136a^4 + 130a^3 + 84a^2 + 39a + 61 + (25a^4 + 60a^3 + 40a^2 + 47a + 6)139 + (21a^4 + 13a^3 + 73a^2 + 64a + 8)139^2 + (40a^4 + 26a^3 + 122a^2 + 89a + 96)139^3 + (20a^4 + 102a^3 + 118a^2 + 107a + 26)139^4+O(139^5)
$r_{ 15 }$	$=$	$ 138 a^{4} + 92 a^{3} + 84 a^{2} + 87 a + 108 + \left(18 a^{4} + 35 a^{3} + 78 a^{2} + 68 a + 61\right)\cdot 139 + \left(2 a^{4} + 121 a^{3} + 14 a^{2} + 21 a + 19\right)\cdot 139^{2} + \left(83 a^{4} + 6 a^{3} + 70 a^{2} + 25 a + 12\right)\cdot 139^{3} + \left(7 a^{4} + 118 a^{3} + 79 a^{2} + 4 a + 5\right)\cdot 139^{4} +O(139^{5})$ 138a^4 + 92a^3 + 84a^2 + 87a + 108 + (18a^4 + 35a^3 + 78a^2 + 68a + 61)139 + (2a^4 + 121a^3 + 14a^2 + 21a + 19)139^2 + (83a^4 + 6a^3 + 70a^2 + 25a + 12)139^3 + (7a^4 + 118a^3 + 79a^2 + 4a + 5)139^4+O(139^5)

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

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

Character values on conjugacy classes

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

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	2.1452.15t4.a.c
Dimension	$2$
Group	$S_3 \times C_5$
Conductor	$1452$
Root number	not computed
Indicator	$0$

$r_{ 1 }$	$=$	\( 18 a^{4} + 13 a^{3} + 54 a^{2} + 42 a + 121 + \left(3 a^{4} + 29 a^{3} + 90 a^{2} + 71 a + 73\right)\cdot 139 + \left(24 a^{4} + 82 a^{3} + 56 a^{2} + 112 a + 55\right)\cdot 139^{2} + \left(120 a^{4} + 44 a^{3} + 26 a^{2} + 13 a + 31\right)\cdot 139^{3} + \left(43 a^{4} + 35 a^{3} + 120 a^{2} + 73 a + 17\right)\cdot 139^{4} +O(139^{5})\) 18a^4 + 13a^3 + 54a^2 + 42a + 121 + (3a^4 + 29a^3 + 90a^2 + 71a + 73)139 + (24a^4 + 82a^3 + 56a^2 + 112a + 55)139^2 + (120a^4 + 44a^3 + 26a^2 + 13a + 31)139^3 + (43a^4 + 35a^3 + 120a^2 + 73a + 17)139^4+O(139^5)
$r_{ 2 }$	$=$	\( 27 a^{4} + 68 a^{3} + 75 a^{2} + 54 a + 23 + \left(121 a^{4} + 27 a^{3} + 34 a^{2} + 74 a + 73\right)\cdot 139 + \left(49 a^{4} + 88 a^{3} + 32 a^{2} + 134 a + 98\right)\cdot 139^{2} + \left(43 a^{4} + 120 a^{3} + 55 a^{2} + 18 a + 121\right)\cdot 139^{3} + \left(22 a^{4} + 91 a^{3} + 136 a^{2} + 23 a + 42\right)\cdot 139^{4} +O(139^{5})\) 27a^4 + 68a^3 + 75a^2 + 54a + 23 + (121a^4 + 27a^3 + 34a^2 + 74a + 73)139 + (49a^4 + 88a^3 + 32a^2 + 134a + 98)139^2 + (43a^4 + 120a^3 + 55a^2 + 18a + 121)139^3 + (22a^4 + 91a^3 + 136a^2 + 23a + 42)139^4+O(139^5)
$r_{ 3 }$	$=$	\( 33 a^{4} + 13 a^{3} + 114 a^{2} + 138 a + 102 + \left(53 a^{4} + 25 a^{3} + 72 a^{2} + 31 a + 57\right)\cdot 139 + \left(67 a^{4} + 10 a^{3} + 44 a^{2} + 44 a + 124\right)\cdot 139^{2} + \left(44 a^{4} + 125 a^{3} + 127 a^{2} + 83 a + 120\right)\cdot 139^{3} + \left(128 a^{4} + 102 a^{3} + 3 a^{2} + 62 a + 136\right)\cdot 139^{4} +O(139^{5})\) 33a^4 + 13a^3 + 114a^2 + 138a + 102 + (53a^4 + 25a^3 + 72a^2 + 31a + 57)139 + (67a^4 + 10a^3 + 44a^2 + 44a + 124)139^2 + (44a^4 + 125a^3 + 127a^2 + 83a + 120)139^3 + (128a^4 + 102a^3 + 3a^2 + 62a + 136)139^4+O(139^5)
$r_{ 4 }$	$=$	\( 33 a^{4} + 125 a^{3} + 72 a^{2} + 106 a + 64 + \left(68 a^{4} + 129 a^{3} + 28 a^{2} + 121 a + 3\right)\cdot 139 + \left(17 a^{4} + 28 a^{3} + 110 a^{2} + 11 a + 21\right)\cdot 139^{2} + \left(83 a^{4} + 78 a^{3} + 9 a^{2} + 11 a + 13\right)\cdot 139^{3} + \left(70 a^{4} + 134 a^{3} + 40 a^{2} + 102 a + 61\right)\cdot 139^{4} +O(139^{5})\) 33a^4 + 125a^3 + 72a^2 + 106a + 64 + (68a^4 + 129a^3 + 28a^2 + 121a + 3)139 + (17a^4 + 28a^3 + 110a^2 + 11a + 21)139^2 + (83a^4 + 78a^3 + 9a^2 + 11a + 13)139^3 + (70a^4 + 134a^3 + 40a^2 + 102a + 61)139^4+O(139^5)
$r_{ 5 }$	$=$	\( 45 a^{4} + 134 a^{3} + 59 a^{2} + 93 a + 28 + \left(66 a^{4} + 78 a^{3} + 92 a^{2} + 109 a + 51\right)\cdot 139 + \left(77 a^{4} + 120 a^{3} + 39 a^{2} + 39 a + 41\right)\cdot 139^{2} + \left(137 a^{4} + 37 a^{3} + 119 a^{2} + 48 a + 41\right)\cdot 139^{3} + \left(102 a^{4} + 83 a^{3} + 134 a^{2} + 132 a + 132\right)\cdot 139^{4} +O(139^{5})\) 45a^4 + 134a^3 + 59a^2 + 93a + 28 + (66a^4 + 78a^3 + 92a^2 + 109a + 51)139 + (77a^4 + 120a^3 + 39a^2 + 39a + 41)139^2 + (137a^4 + 37a^3 + 119a^2 + 48a + 41)139^3 + (102a^4 + 83a^3 + 134a^2 + 132a + 132)139^4+O(139^5)
$r_{ 6 }$	$=$	\( 79 a^{4} + 20 a^{3} + 27 a^{2} + 87 a + 15 + \left(119 a^{4} + 97 a^{3} + 102 a^{2} + 60 a + 136\right)\cdot 139 + \left(41 a^{4} + 25 a^{3} + 68 a^{2} + 132 a + 76\right)\cdot 139^{2} + \left(102 a^{4} + 92 a^{3} + 49 a^{2} + 14 a + 27\right)\cdot 139^{3} + \left(11 a^{4} + 51 a^{3} + 84 a^{2} + 59 a + 7\right)\cdot 139^{4} +O(139^{5})\) 79a^4 + 20a^3 + 27a^2 + 87a + 15 + (119a^4 + 97a^3 + 102a^2 + 60a + 136)139 + (41a^4 + 25a^3 + 68a^2 + 132a + 76)139^2 + (102a^4 + 92a^3 + 49a^2 + 14a + 27)139^3 + (11a^4 + 51a^3 + 84a^2 + 59a + 7)139^4+O(139^5)
$r_{ 7 }$	$=$	\( 85 a^{4} + 113 a^{3} + 103 a^{2} + 115 a + 63 + \left(118 a^{4} + 112 a^{3} + 46 a^{2} + 96 a + 128\right)\cdot 139 + \left(125 a^{4} + 90 a^{3} + 80 a^{2} + 54 a + 53\right)\cdot 139^{2} + \left(81 a^{4} + 14 a^{3} + 14 a^{2} + 72 a + 3\right)\cdot 139^{3} + \left(44 a^{4} + 110 a^{3} + 67 a^{2} + 136 a + 131\right)\cdot 139^{4} +O(139^{5})\) 85a^4 + 113a^3 + 103a^2 + 115a + 63 + (118a^4 + 112a^3 + 46a^2 + 96a + 128)139 + (125a^4 + 90a^3 + 80a^2 + 54a + 53)139^2 + (81a^4 + 14a^3 + 14a^2 + 72a + 3)139^3 + (44a^4 + 110a^3 + 67a^2 + 136a + 131)139^4+O(139^5)
$r_{ 8 }$	$=$	\( 91 a^{4} + 23 a^{3} + 135 a^{2} + 119 a + 118 + \left(24 a^{4} + 68 a^{3} + 91 a^{2} + 110 a + 134\right)\cdot 139 + \left(103 a^{4} + 111 a^{3} + 138 a^{2} + 102 a + 107\right)\cdot 139^{2} + \left(28 a^{4} + 20 a^{3} + 65 a^{2} + 92 a + 4\right)\cdot 139^{3} + \left(29 a^{4} + 42 a^{3} + 31 a^{2} + 50 a + 98\right)\cdot 139^{4} +O(139^{5})\) 91a^4 + 23a^3 + 135a^2 + 119a + 118 + (24a^4 + 68a^3 + 91a^2 + 110a + 134)139 + (103a^4 + 111a^3 + 138a^2 + 102a + 107)139^2 + (28a^4 + 20a^3 + 65a^2 + 92a + 4)139^3 + (29a^4 + 42a^3 + 31a^2 + 50a + 98)139^4+O(139^5)
$r_{ 9 }$	$=$	\( 100 a^{4} + 134 a^{3} + 120 a + 44 + \left(47 a^{4} + 10 a^{3} + 84 a^{2} + 74 a + 117\right)\cdot 139 + \left(59 a^{4} + 114 a^{3} + 19 a^{2} + 134 a + 77\right)\cdot 139^{2} + \left(48 a^{4} + 108 a^{3} + 17 a^{2} + 94 a + 13\right)\cdot 139^{3} + \left(87 a^{4} + 95 a^{3} + 6 a^{2} + 38 a + 56\right)\cdot 139^{4} +O(139^{5})\) 100a^4 + 134a^3 + 120a + 44 + (47a^4 + 10a^3 + 84a^2 + 74a + 117)139 + (59a^4 + 114a^3 + 19a^2 + 134a + 77)139^2 + (48a^4 + 108a^3 + 17a^2 + 94a + 13)139^3 + (87a^4 + 95a^3 + 6a^2 + 38a + 56)*139^4+O(139^5)
$r_{ 10 }$	$=$	\( 111 a^{4} + 26 a^{3} + 100 a^{2} + 35 a + 31 + \left(5 a^{4} + 62 a^{3} + 39 a^{2} + 132 a + 95\right)\cdot 139 + \left(118 a^{4} + 74 a^{3} + 54 a^{2} + 38 a + 112\right)\cdot 139^{2} + \left(38 a^{4} + 108 a^{3} + 54 a^{2} + 88 a + 75\right)\cdot 139^{3} + \left(36 a^{4} + 81 a^{3} + 86 a^{2} + 59 a + 95\right)\cdot 139^{4} +O(139^{5})\) 111a^4 + 26a^3 + 100a^2 + 35a + 31 + (5a^4 + 62a^3 + 39a^2 + 132a + 95)139 + (118a^4 + 74a^3 + 54a^2 + 38a + 112)139^2 + (38a^4 + 108a^3 + 54a^2 + 88a + 75)139^3 + (36a^4 + 81a^3 + 86a^2 + 59a + 95)139^4+O(139^5)
$r_{ 11 }$	$=$	\( 117 a^{4} + 134 a^{3} + 65 a^{2} + 115 a + 79 + \left(57 a^{4} + 125 a^{3} + 135 a^{2} + 112 a + 94\right)\cdot 139 + \left(66 a^{4} + 128 a^{3} + 107 a^{2} + 60 a + 116\right)\cdot 139^{2} + \left(130 a^{4} + 131 a^{3} + 138 a^{2} + 67 a + 113\right)\cdot 139^{3} + \left(61 a^{4} + 78 a^{3} + 126 a^{2} + 78 a + 22\right)\cdot 139^{4} +O(139^{5})\) 117a^4 + 134a^3 + 65a^2 + 115a + 79 + (57a^4 + 125a^3 + 135a^2 + 112a + 94)139 + (66a^4 + 128a^3 + 107a^2 + 60a + 116)139^2 + (130a^4 + 131a^3 + 138a^2 + 67a + 113)139^3 + (61a^4 + 78a^3 + 126a^2 + 78a + 22)139^4+O(139^5)
$r_{ 12 }$	$=$	\( 118 a^{4} + 62 a^{3} + 64 a^{2} + 112 a + 56 + \left(39 a^{4} + 42 a^{3} + 18 a^{2} + 74 a + 117\right)\cdot 139 + \left(26 a^{4} + 83 a^{3} + 133 a^{2} + 75 a + 48\right)\cdot 139^{2} + \left(28 a^{4} + 72 a^{3} + 53 a^{2} + 28 a\right)\cdot 139^{3} + \left(103 a^{4} + 97 a^{3} + 134 a^{2} + 103 a + 134\right)\cdot 139^{4} +O(139^{5})\) 118a^4 + 62a^3 + 64a^2 + 112a + 56 + (39a^4 + 42a^3 + 18a^2 + 74a + 117)139 + (26a^4 + 83a^3 + 133a^2 + 75a + 48)139^2 + (28a^4 + 72a^3 + 53a^2 + 28a)139^3 + (103a^4 + 97a^3 + 134a^2 + 103a + 134)139^4+O(139^5)
$r_{ 13 }$	$=$	\( 120 a^{4} + 25 a^{3} + 76 a^{2} + 128 a + 65 + \left(62 a^{4} + 66 a^{3} + 16 a^{2} + 62 a + 99\right)\cdot 139 + \left(33 a^{4} + 18 a^{3} + 138 a^{2} + 83 a + 9\right)\cdot 139^{2} + \left(101 a^{4} + 123 a^{3} + 47 a^{2} + 84 a + 19\right)\cdot 139^{3} + \left(63 a^{4} + 24 a^{3} + 80 a^{2} + 80 a + 6\right)\cdot 139^{4} +O(139^{5})\) 120a^4 + 25a^3 + 76a^2 + 128a + 65 + (62a^4 + 66a^3 + 16a^2 + 62a + 99)139 + (33a^4 + 18a^3 + 138a^2 + 83a + 9)139^2 + (101a^4 + 123a^3 + 47a^2 + 84a + 19)139^3 + (63a^4 + 24a^3 + 80a^2 + 80a + 6)139^4+O(139^5)
$r_{ 14 }$	$=$	\( 136 a^{4} + 130 a^{3} + 84 a^{2} + 39 a + 61 + \left(25 a^{4} + 60 a^{3} + 40 a^{2} + 47 a + 6\right)\cdot 139 + \left(21 a^{4} + 13 a^{3} + 73 a^{2} + 64 a + 8\right)\cdot 139^{2} + \left(40 a^{4} + 26 a^{3} + 122 a^{2} + 89 a + 96\right)\cdot 139^{3} + \left(20 a^{4} + 102 a^{3} + 118 a^{2} + 107 a + 26\right)\cdot 139^{4} +O(139^{5})\) 136a^4 + 130a^3 + 84a^2 + 39a + 61 + (25a^4 + 60a^3 + 40a^2 + 47a + 6)139 + (21a^4 + 13a^3 + 73a^2 + 64a + 8)139^2 + (40a^4 + 26a^3 + 122a^2 + 89a + 96)139^3 + (20a^4 + 102a^3 + 118a^2 + 107a + 26)139^4+O(139^5)
$r_{ 15 }$	$=$	\( 138 a^{4} + 92 a^{3} + 84 a^{2} + 87 a + 108 + \left(18 a^{4} + 35 a^{3} + 78 a^{2} + 68 a + 61\right)\cdot 139 + \left(2 a^{4} + 121 a^{3} + 14 a^{2} + 21 a + 19\right)\cdot 139^{2} + \left(83 a^{4} + 6 a^{3} + 70 a^{2} + 25 a + 12\right)\cdot 139^{3} + \left(7 a^{4} + 118 a^{3} + 79 a^{2} + 4 a + 5\right)\cdot 139^{4} +O(139^{5})\) 138a^4 + 92a^3 + 84a^2 + 87a + 108 + (18a^4 + 35a^3 + 78a^2 + 68a + 61)139 + (2a^4 + 121a^3 + 14a^2 + 21a + 19)139^2 + (83a^4 + 6a^3 + 70a^2 + 25a + 12)139^3 + (7a^4 + 118a^3 + 79a^2 + 4a + 5)139^4+O(139^5)