LMFDB - Artin representation 2.239.15t2.a.b

Basic invariants

Dimension:	$2$
Group:	$D_{15}$
Conductor:	$239$
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin stem field:	Galois closure of 15.1.44543599279432079.1
Galois orbit size:	$4$
Smallest permutation container:	$D_{15}$
Parity:	odd
Determinant:	1.239.2t1.a.a
Projective image:	$D_{15}$
Projective stem field:	Galois closure of 15.1.44543599279432079.1

Defining polynomial

$f(x)$

$=$

$ x^{15} - 4 x^{14} + 4 x^{13} + 4 x^{12} - 5 x^{11} - 13 x^{10} + 20 x^{9} + 4 x^{8} - 15 x^{7} + \cdots - 1 $

The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$: $ x^{5} + 18x + 64 $ x^5 + 18*x + 64

Roots:

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

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

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

Character values on conjugacy classes

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

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	2.239.15t2.a.b
Dimension	$2$
Group	$D_{15}$
Conductor	$239$
Root number	$1$
Indicator	$1$

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