LMFDB - Artin representation 10.267403334175880281849856.20t77.a.a

Basic invariants

Dimension:	$10$
Group:	$(C_2^4 : C_5):C_4$
Conductor:	$267\!\cdots\!856$$\medspace = 2^{32} \cdot 53^{8} $
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin stem field:	Galois closure of 10.10.23241017135202304.1
Galois orbit size:	$1$
Smallest permutation container:	20T77
Parity:	even
Determinant:	1.1.1t1.a.a
Projective image:	$C_2^4:F_5$
Projective stem field:	Galois closure of 10.10.23241017135202304.1

Defining polynomial

$f(x)$

$=$

$ x^{10} - 13x^{8} + 56x^{6} - 90x^{4} + 40x^{2} - 4 $

The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: $ x^{4} + 3x^{2} + 40x + 2 $ x^4 + 3*x^2 + 40*x + 2

Roots:

$r_{ 1 }$	$=$	$ 36 + 38\cdot 61 + 55\cdot 61^{2} + 32\cdot 61^{3} + 8\cdot 61^{4} + 57\cdot 61^{5} + 34\cdot 61^{6} + 48\cdot 61^{7} + 40\cdot 61^{8} + 40\cdot 61^{9} +O(61^{10})$ 36 + 3861 + 5561^2 + 3261^3 + 861^4 + 5761^5 + 3461^6 + 4861^7 + 4061^8 + 40*61^9+O(61^10)
$r_{ 2 }$	$=$	$ 25 + 22\cdot 61 + 5\cdot 61^{2} + 28\cdot 61^{3} + 52\cdot 61^{4} + 3\cdot 61^{5} + 26\cdot 61^{6} + 12\cdot 61^{7} + 20\cdot 61^{8} + 20\cdot 61^{9} +O(61^{10})$ 25 + 2261 + 561^2 + 2861^3 + 5261^4 + 361^5 + 2661^6 + 1261^7 + 2061^8 + 20*61^9+O(61^10)
$r_{ 3 }$	$=$	$ 21 a^{3} + 7 a^{2} + 35 a + 1 + \left(13 a^{3} + 46 a^{2} + 2 a + 26\right)\cdot 61 + \left(21 a^{3} + 5 a^{2} + 52 a + 10\right)\cdot 61^{2} + \left(53 a^{3} + 49 a^{2} + 5 a + 9\right)\cdot 61^{3} + \left(55 a^{3} + 29 a^{2} + 19 a + 57\right)\cdot 61^{4} + \left(27 a^{3} + 43 a^{2} + a + 22\right)\cdot 61^{5} + \left(22 a^{3} + 4 a^{2} + 60 a + 42\right)\cdot 61^{6} + \left(6 a^{3} + 20 a^{2} + 35 a + 21\right)\cdot 61^{7} + \left(54 a^{3} + 22 a^{2} + 3\right)\cdot 61^{8} + \left(50 a^{3} + 54 a^{2} + 33 a + 46\right)\cdot 61^{9} +O(61^{10})$ 21a^3 + 7a^2 + 35a + 1 + (13a^3 + 46a^2 + 2a + 26)61 + (21a^3 + 5a^2 + 52a + 10)61^2 + (53a^3 + 49a^2 + 5a + 9)61^3 + (55a^3 + 29a^2 + 19a + 57)61^4 + (27a^3 + 43a^2 + a + 22)61^5 + (22a^3 + 4a^2 + 60a + 42)61^6 + (6a^3 + 20a^2 + 35a + 21)61^7 + (54a^3 + 22a^2 + 3)61^8 + (50a^3 + 54a^2 + 33a + 46)*61^9+O(61^10)
$r_{ 4 }$	$=$	$ 56 a^{3} + 23 a^{2} + 31 a + 36 + \left(20 a^{3} + a^{2} + 18 a + 5\right)\cdot 61 + \left(37 a^{3} + 36 a^{2} + 40 a + 40\right)\cdot 61^{2} + \left(45 a^{3} + 15 a^{2} + 34 a + 6\right)\cdot 61^{3} + \left(32 a^{3} + 10 a^{2} + 17 a + 8\right)\cdot 61^{4} + \left(48 a^{3} + 56 a^{2} + 59 a + 41\right)\cdot 61^{5} + \left(5 a^{3} + 20 a^{2} + 55 a + 50\right)\cdot 61^{6} + \left(17 a^{3} + 21 a^{2} + 4 a + 42\right)\cdot 61^{7} + \left(53 a^{3} + 40 a^{2} + 45 a + 48\right)\cdot 61^{8} + \left(54 a^{3} + 60 a^{2} + 46 a + 35\right)\cdot 61^{9} +O(61^{10})$ 56a^3 + 23a^2 + 31a + 36 + (20a^3 + a^2 + 18a + 5)61 + (37a^3 + 36a^2 + 40a + 40)61^2 + (45a^3 + 15a^2 + 34a + 6)61^3 + (32a^3 + 10a^2 + 17a + 8)61^4 + (48a^3 + 56a^2 + 59a + 41)61^5 + (5a^3 + 20a^2 + 55a + 50)61^6 + (17a^3 + 21a^2 + 4a + 42)61^7 + (53a^3 + 40a^2 + 45a + 48)61^8 + (54a^3 + 60a^2 + 46a + 35)61^9+O(61^10)
$r_{ 5 }$	$=$	$ 51 a^{3} + 19 a^{2} + 22 a + 2 + \left(37 a^{3} + 50 a^{2} + 47 a + 7\right)\cdot 61 + \left(41 a^{3} + 37 a^{2} + 29 a + 49\right)\cdot 61^{2} + \left(43 a^{3} + 33 a^{2} + 26 a + 36\right)\cdot 61^{3} + \left(54 a^{3} + 25 a^{2} + 18 a + 49\right)\cdot 61^{4} + \left(30 a^{3} + 59 a^{2} + 58 a + 4\right)\cdot 61^{5} + \left(56 a^{3} + 20 a^{2} + 52 a + 47\right)\cdot 61^{6} + \left(25 a^{3} + 40 a^{2} + 8 a\right)\cdot 61^{7} + \left(4 a^{3} + 48 a^{2} + 60 a + 29\right)\cdot 61^{8} + \left(23 a^{3} + 60 a^{2} + 41 a + 27\right)\cdot 61^{9} +O(61^{10})$ 51a^3 + 19a^2 + 22a + 2 + (37a^3 + 50a^2 + 47a + 7)61 + (41a^3 + 37a^2 + 29a + 49)61^2 + (43a^3 + 33a^2 + 26a + 36)61^3 + (54a^3 + 25a^2 + 18a + 49)61^4 + (30a^3 + 59a^2 + 58a + 4)61^5 + (56a^3 + 20a^2 + 52a + 47)61^6 + (25a^3 + 40a^2 + 8a)61^7 + (4a^3 + 48a^2 + 60a + 29)61^8 + (23a^3 + 60a^2 + 41a + 27)*61^9+O(61^10)
$r_{ 6 }$	$=$	$ 25 a^{3} + 35 a^{2} + 18 a + 41 + \left(45 a^{3} + 5 a^{2} + 2 a + 42\right)\cdot 61 + \left(57 a^{3} + 7 a^{2} + 18 a + 40\right)\cdot 61^{2} + \left(35 a^{3} + 55 a + 23\right)\cdot 61^{3} + \left(31 a^{3} + 6 a^{2} + 16 a + 55\right)\cdot 61^{4} + \left(51 a^{3} + 11 a^{2} + 55 a + 41\right)\cdot 61^{5} + \left(39 a^{3} + 37 a^{2} + 48 a + 2\right)\cdot 61^{6} + \left(36 a^{3} + 41 a^{2} + 38 a + 17\right)\cdot 61^{7} + \left(3 a^{3} + 5 a^{2} + 43 a + 49\right)\cdot 61^{8} + \left(27 a^{3} + 6 a^{2} + 55 a + 51\right)\cdot 61^{9} +O(61^{10})$ 25a^3 + 35a^2 + 18a + 41 + (45a^3 + 5a^2 + 2a + 42)61 + (57a^3 + 7a^2 + 18a + 40)61^2 + (35a^3 + 55a + 23)61^3 + (31a^3 + 6a^2 + 16a + 55)61^4 + (51a^3 + 11a^2 + 55a + 41)61^5 + (39a^3 + 37a^2 + 48a + 2)61^6 + (36a^3 + 41a^2 + 38a + 17)61^7 + (3a^3 + 5a^2 + 43a + 49)61^8 + (27a^3 + 6a^2 + 55a + 51)61^9+O(61^10)
$r_{ 7 }$	$=$	$ 40 a^{3} + 54 a^{2} + 26 a + 60 + \left(47 a^{3} + 14 a^{2} + 58 a + 34\right)\cdot 61 + \left(39 a^{3} + 55 a^{2} + 8 a + 50\right)\cdot 61^{2} + \left(7 a^{3} + 11 a^{2} + 55 a + 51\right)\cdot 61^{3} + \left(5 a^{3} + 31 a^{2} + 41 a + 3\right)\cdot 61^{4} + \left(33 a^{3} + 17 a^{2} + 59 a + 38\right)\cdot 61^{5} + \left(38 a^{3} + 56 a^{2} + 18\right)\cdot 61^{6} + \left(54 a^{3} + 40 a^{2} + 25 a + 39\right)\cdot 61^{7} + \left(6 a^{3} + 38 a^{2} + 60 a + 57\right)\cdot 61^{8} + \left(10 a^{3} + 6 a^{2} + 27 a + 14\right)\cdot 61^{9} +O(61^{10})$ 40a^3 + 54a^2 + 26a + 60 + (47a^3 + 14a^2 + 58a + 34)61 + (39a^3 + 55a^2 + 8a + 50)61^2 + (7a^3 + 11a^2 + 55a + 51)61^3 + (5a^3 + 31a^2 + 41a + 3)61^4 + (33a^3 + 17a^2 + 59a + 38)61^5 + (38a^3 + 56a^2 + 18)61^6 + (54a^3 + 40a^2 + 25a + 39)61^7 + (6a^3 + 38a^2 + 60a + 57)61^8 + (10a^3 + 6a^2 + 27a + 14)61^9+O(61^10)
$r_{ 8 }$	$=$	$ 5 a^{3} + 38 a^{2} + 30 a + 25 + \left(40 a^{3} + 59 a^{2} + 42 a + 55\right)\cdot 61 + \left(23 a^{3} + 24 a^{2} + 20 a + 20\right)\cdot 61^{2} + \left(15 a^{3} + 45 a^{2} + 26 a + 54\right)\cdot 61^{3} + \left(28 a^{3} + 50 a^{2} + 43 a + 52\right)\cdot 61^{4} + \left(12 a^{3} + 4 a^{2} + a + 19\right)\cdot 61^{5} + \left(55 a^{3} + 40 a^{2} + 5 a + 10\right)\cdot 61^{6} + \left(43 a^{3} + 39 a^{2} + 56 a + 18\right)\cdot 61^{7} + \left(7 a^{3} + 20 a^{2} + 15 a + 12\right)\cdot 61^{8} + \left(6 a^{3} + 14 a + 25\right)\cdot 61^{9} +O(61^{10})$ 5a^3 + 38a^2 + 30a + 25 + (40a^3 + 59a^2 + 42a + 55)61 + (23a^3 + 24a^2 + 20a + 20)61^2 + (15a^3 + 45a^2 + 26a + 54)61^3 + (28a^3 + 50a^2 + 43a + 52)61^4 + (12a^3 + 4a^2 + a + 19)61^5 + (55a^3 + 40a^2 + 5a + 10)61^6 + (43a^3 + 39a^2 + 56a + 18)61^7 + (7a^3 + 20a^2 + 15a + 12)61^8 + (6a^3 + 14a + 25)*61^9+O(61^10)
$r_{ 9 }$	$=$	$ 36 a^{3} + 26 a^{2} + 43 a + 20 + \left(15 a^{3} + 55 a^{2} + 58 a + 18\right)\cdot 61 + \left(3 a^{3} + 53 a^{2} + 42 a + 20\right)\cdot 61^{2} + \left(25 a^{3} + 60 a^{2} + 5 a + 37\right)\cdot 61^{3} + \left(29 a^{3} + 54 a^{2} + 44 a + 5\right)\cdot 61^{4} + \left(9 a^{3} + 49 a^{2} + 5 a + 19\right)\cdot 61^{5} + \left(21 a^{3} + 23 a^{2} + 12 a + 58\right)\cdot 61^{6} + \left(24 a^{3} + 19 a^{2} + 22 a + 43\right)\cdot 61^{7} + \left(57 a^{3} + 55 a^{2} + 17 a + 11\right)\cdot 61^{8} + \left(33 a^{3} + 54 a^{2} + 5 a + 9\right)\cdot 61^{9} +O(61^{10})$ 36a^3 + 26a^2 + 43a + 20 + (15a^3 + 55a^2 + 58a + 18)61 + (3a^3 + 53a^2 + 42a + 20)61^2 + (25a^3 + 60a^2 + 5a + 37)61^3 + (29a^3 + 54a^2 + 44a + 5)61^4 + (9a^3 + 49a^2 + 5a + 19)61^5 + (21a^3 + 23a^2 + 12a + 58)61^6 + (24a^3 + 19a^2 + 22a + 43)61^7 + (57a^3 + 55a^2 + 17a + 11)61^8 + (33a^3 + 54a^2 + 5a + 9)*61^9+O(61^10)
$r_{ 10 }$	$=$	$ 10 a^{3} + 42 a^{2} + 39 a + 59 + \left(23 a^{3} + 10 a^{2} + 13 a + 53\right)\cdot 61 + \left(19 a^{3} + 23 a^{2} + 31 a + 11\right)\cdot 61^{2} + \left(17 a^{3} + 27 a^{2} + 34 a + 24\right)\cdot 61^{3} + \left(6 a^{3} + 35 a^{2} + 42 a + 11\right)\cdot 61^{4} + \left(30 a^{3} + a^{2} + 2 a + 56\right)\cdot 61^{5} + \left(4 a^{3} + 40 a^{2} + 8 a + 13\right)\cdot 61^{6} + \left(35 a^{3} + 20 a^{2} + 52 a + 60\right)\cdot 61^{7} + \left(56 a^{3} + 12 a^{2} + 31\right)\cdot 61^{8} + \left(37 a^{3} + 19 a + 33\right)\cdot 61^{9} +O(61^{10})$ 10a^3 + 42a^2 + 39a + 59 + (23a^3 + 10a^2 + 13a + 53)61 + (19a^3 + 23a^2 + 31a + 11)61^2 + (17a^3 + 27a^2 + 34a + 24)61^3 + (6a^3 + 35a^2 + 42a + 11)61^4 + (30a^3 + a^2 + 2a + 56)61^5 + (4a^3 + 40a^2 + 8a + 13)61^6 + (35a^3 + 20a^2 + 52a + 60)61^7 + (56a^3 + 12a^2 + 31)61^8 + (37a^3 + 19a + 33)61^9+O(61^10)

Generators of the action on the roots $r_1, \ldots, r_{ 10 }$

Cycle notation
$(1,5,2,10)(3,9,7,6)$
$(1,9,5,7,2,6,10,3)(4,8)$
$(1,4,5,6,3)(2,8,10,9,7)$
$(1,2)(6,9)$
$(3,7)(6,9)$
$(5,10)(6,9)$
$(4,8)(6,9)$

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 10 }$	Character value
$1$	$1$	$()$	$10$
$5$	$2$	$(1,2)(3,7)(5,10)(6,9)$	$2$
$10$	$2$	$(1,2)(6,9)$	$-2$
$20$	$2$	$(3,8)(4,7)(5,9)(6,10)$	$-2$
$20$	$4$	$(1,5,2,10)(3,9,7,6)$	$2$
$40$	$4$	$(1,6)(2,9)(3,7)(4,5,8,10)$	$0$
$40$	$4$	$(1,9,7,8)(2,6,3,4)$	$0$
$40$	$4$	$(1,8,7,9)(2,4,3,6)$	$0$
$64$	$5$	$(1,4,5,6,3)(2,8,10,9,7)$	$0$
$40$	$8$	$(1,9,5,7,2,6,10,3)(4,8)$	$0$
$40$	$8$	$(1,7,10,9,2,3,5,6)(4,8)$	$0$

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 $r_1, \ldots, r_{ 10 }$

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	10.267...856.20t77.a.a
Dimension	$10$
Group	$(C_2^4 : C_5):C_4$
Conductor	$2.674\times 10^{23}$
Root number	$1$
Indicator	$1$

$r_{ 1 }$	$=$	\( 36 + 38\cdot 61 + 55\cdot 61^{2} + 32\cdot 61^{3} + 8\cdot 61^{4} + 57\cdot 61^{5} + 34\cdot 61^{6} + 48\cdot 61^{7} + 40\cdot 61^{8} + 40\cdot 61^{9} +O(61^{10})\) 36 + 3861 + 5561^2 + 3261^3 + 861^4 + 5761^5 + 3461^6 + 4861^7 + 4061^8 + 40*61^9+O(61^10)
$r_{ 2 }$	$=$	\( 25 + 22\cdot 61 + 5\cdot 61^{2} + 28\cdot 61^{3} + 52\cdot 61^{4} + 3\cdot 61^{5} + 26\cdot 61^{6} + 12\cdot 61^{7} + 20\cdot 61^{8} + 20\cdot 61^{9} +O(61^{10})\) 25 + 2261 + 561^2 + 2861^3 + 5261^4 + 361^5 + 2661^6 + 1261^7 + 2061^8 + 20*61^9+O(61^10)
$r_{ 3 }$	$=$	\( 21 a^{3} + 7 a^{2} + 35 a + 1 + \left(13 a^{3} + 46 a^{2} + 2 a + 26\right)\cdot 61 + \left(21 a^{3} + 5 a^{2} + 52 a + 10\right)\cdot 61^{2} + \left(53 a^{3} + 49 a^{2} + 5 a + 9\right)\cdot 61^{3} + \left(55 a^{3} + 29 a^{2} + 19 a + 57\right)\cdot 61^{4} + \left(27 a^{3} + 43 a^{2} + a + 22\right)\cdot 61^{5} + \left(22 a^{3} + 4 a^{2} + 60 a + 42\right)\cdot 61^{6} + \left(6 a^{3} + 20 a^{2} + 35 a + 21\right)\cdot 61^{7} + \left(54 a^{3} + 22 a^{2} + 3\right)\cdot 61^{8} + \left(50 a^{3} + 54 a^{2} + 33 a + 46\right)\cdot 61^{9} +O(61^{10})\) 21a^3 + 7a^2 + 35a + 1 + (13a^3 + 46a^2 + 2a + 26)61 + (21a^3 + 5a^2 + 52a + 10)61^2 + (53a^3 + 49a^2 + 5a + 9)61^3 + (55a^3 + 29a^2 + 19a + 57)61^4 + (27a^3 + 43a^2 + a + 22)61^5 + (22a^3 + 4a^2 + 60a + 42)61^6 + (6a^3 + 20a^2 + 35a + 21)61^7 + (54a^3 + 22a^2 + 3)61^8 + (50a^3 + 54a^2 + 33a + 46)*61^9+O(61^10)
$r_{ 4 }$	$=$	\( 56 a^{3} + 23 a^{2} + 31 a + 36 + \left(20 a^{3} + a^{2} + 18 a + 5\right)\cdot 61 + \left(37 a^{3} + 36 a^{2} + 40 a + 40\right)\cdot 61^{2} + \left(45 a^{3} + 15 a^{2} + 34 a + 6\right)\cdot 61^{3} + \left(32 a^{3} + 10 a^{2} + 17 a + 8\right)\cdot 61^{4} + \left(48 a^{3} + 56 a^{2} + 59 a + 41\right)\cdot 61^{5} + \left(5 a^{3} + 20 a^{2} + 55 a + 50\right)\cdot 61^{6} + \left(17 a^{3} + 21 a^{2} + 4 a + 42\right)\cdot 61^{7} + \left(53 a^{3} + 40 a^{2} + 45 a + 48\right)\cdot 61^{8} + \left(54 a^{3} + 60 a^{2} + 46 a + 35\right)\cdot 61^{9} +O(61^{10})\) 56a^3 + 23a^2 + 31a + 36 + (20a^3 + a^2 + 18a + 5)61 + (37a^3 + 36a^2 + 40a + 40)61^2 + (45a^3 + 15a^2 + 34a + 6)61^3 + (32a^3 + 10a^2 + 17a + 8)61^4 + (48a^3 + 56a^2 + 59a + 41)61^5 + (5a^3 + 20a^2 + 55a + 50)61^6 + (17a^3 + 21a^2 + 4a + 42)61^7 + (53a^3 + 40a^2 + 45a + 48)61^8 + (54a^3 + 60a^2 + 46a + 35)61^9+O(61^10)
$r_{ 5 }$	$=$	\( 51 a^{3} + 19 a^{2} + 22 a + 2 + \left(37 a^{3} + 50 a^{2} + 47 a + 7\right)\cdot 61 + \left(41 a^{3} + 37 a^{2} + 29 a + 49\right)\cdot 61^{2} + \left(43 a^{3} + 33 a^{2} + 26 a + 36\right)\cdot 61^{3} + \left(54 a^{3} + 25 a^{2} + 18 a + 49\right)\cdot 61^{4} + \left(30 a^{3} + 59 a^{2} + 58 a + 4\right)\cdot 61^{5} + \left(56 a^{3} + 20 a^{2} + 52 a + 47\right)\cdot 61^{6} + \left(25 a^{3} + 40 a^{2} + 8 a\right)\cdot 61^{7} + \left(4 a^{3} + 48 a^{2} + 60 a + 29\right)\cdot 61^{8} + \left(23 a^{3} + 60 a^{2} + 41 a + 27\right)\cdot 61^{9} +O(61^{10})\) 51a^3 + 19a^2 + 22a + 2 + (37a^3 + 50a^2 + 47a + 7)61 + (41a^3 + 37a^2 + 29a + 49)61^2 + (43a^3 + 33a^2 + 26a + 36)61^3 + (54a^3 + 25a^2 + 18a + 49)61^4 + (30a^3 + 59a^2 + 58a + 4)61^5 + (56a^3 + 20a^2 + 52a + 47)61^6 + (25a^3 + 40a^2 + 8a)61^7 + (4a^3 + 48a^2 + 60a + 29)61^8 + (23a^3 + 60a^2 + 41a + 27)*61^9+O(61^10)
$r_{ 6 }$	$=$	\( 25 a^{3} + 35 a^{2} + 18 a + 41 + \left(45 a^{3} + 5 a^{2} + 2 a + 42\right)\cdot 61 + \left(57 a^{3} + 7 a^{2} + 18 a + 40\right)\cdot 61^{2} + \left(35 a^{3} + 55 a + 23\right)\cdot 61^{3} + \left(31 a^{3} + 6 a^{2} + 16 a + 55\right)\cdot 61^{4} + \left(51 a^{3} + 11 a^{2} + 55 a + 41\right)\cdot 61^{5} + \left(39 a^{3} + 37 a^{2} + 48 a + 2\right)\cdot 61^{6} + \left(36 a^{3} + 41 a^{2} + 38 a + 17\right)\cdot 61^{7} + \left(3 a^{3} + 5 a^{2} + 43 a + 49\right)\cdot 61^{8} + \left(27 a^{3} + 6 a^{2} + 55 a + 51\right)\cdot 61^{9} +O(61^{10})\) 25a^3 + 35a^2 + 18a + 41 + (45a^3 + 5a^2 + 2a + 42)61 + (57a^3 + 7a^2 + 18a + 40)61^2 + (35a^3 + 55a + 23)61^3 + (31a^3 + 6a^2 + 16a + 55)61^4 + (51a^3 + 11a^2 + 55a + 41)61^5 + (39a^3 + 37a^2 + 48a + 2)61^6 + (36a^3 + 41a^2 + 38a + 17)61^7 + (3a^3 + 5a^2 + 43a + 49)61^8 + (27a^3 + 6a^2 + 55a + 51)61^9+O(61^10)
$r_{ 7 }$	$=$	\( 40 a^{3} + 54 a^{2} + 26 a + 60 + \left(47 a^{3} + 14 a^{2} + 58 a + 34\right)\cdot 61 + \left(39 a^{3} + 55 a^{2} + 8 a + 50\right)\cdot 61^{2} + \left(7 a^{3} + 11 a^{2} + 55 a + 51\right)\cdot 61^{3} + \left(5 a^{3} + 31 a^{2} + 41 a + 3\right)\cdot 61^{4} + \left(33 a^{3} + 17 a^{2} + 59 a + 38\right)\cdot 61^{5} + \left(38 a^{3} + 56 a^{2} + 18\right)\cdot 61^{6} + \left(54 a^{3} + 40 a^{2} + 25 a + 39\right)\cdot 61^{7} + \left(6 a^{3} + 38 a^{2} + 60 a + 57\right)\cdot 61^{8} + \left(10 a^{3} + 6 a^{2} + 27 a + 14\right)\cdot 61^{9} +O(61^{10})\) 40a^3 + 54a^2 + 26a + 60 + (47a^3 + 14a^2 + 58a + 34)61 + (39a^3 + 55a^2 + 8a + 50)61^2 + (7a^3 + 11a^2 + 55a + 51)61^3 + (5a^3 + 31a^2 + 41a + 3)61^4 + (33a^3 + 17a^2 + 59a + 38)61^5 + (38a^3 + 56a^2 + 18)61^6 + (54a^3 + 40a^2 + 25a + 39)61^7 + (6a^3 + 38a^2 + 60a + 57)61^8 + (10a^3 + 6a^2 + 27a + 14)61^9+O(61^10)
$r_{ 8 }$	$=$	\( 5 a^{3} + 38 a^{2} + 30 a + 25 + \left(40 a^{3} + 59 a^{2} + 42 a + 55\right)\cdot 61 + \left(23 a^{3} + 24 a^{2} + 20 a + 20\right)\cdot 61^{2} + \left(15 a^{3} + 45 a^{2} + 26 a + 54\right)\cdot 61^{3} + \left(28 a^{3} + 50 a^{2} + 43 a + 52\right)\cdot 61^{4} + \left(12 a^{3} + 4 a^{2} + a + 19\right)\cdot 61^{5} + \left(55 a^{3} + 40 a^{2} + 5 a + 10\right)\cdot 61^{6} + \left(43 a^{3} + 39 a^{2} + 56 a + 18\right)\cdot 61^{7} + \left(7 a^{3} + 20 a^{2} + 15 a + 12\right)\cdot 61^{8} + \left(6 a^{3} + 14 a + 25\right)\cdot 61^{9} +O(61^{10})\) 5a^3 + 38a^2 + 30a + 25 + (40a^3 + 59a^2 + 42a + 55)61 + (23a^3 + 24a^2 + 20a + 20)61^2 + (15a^3 + 45a^2 + 26a + 54)61^3 + (28a^3 + 50a^2 + 43a + 52)61^4 + (12a^3 + 4a^2 + a + 19)61^5 + (55a^3 + 40a^2 + 5a + 10)61^6 + (43a^3 + 39a^2 + 56a + 18)61^7 + (7a^3 + 20a^2 + 15a + 12)61^8 + (6a^3 + 14a + 25)*61^9+O(61^10)
$r_{ 9 }$	$=$	\( 36 a^{3} + 26 a^{2} + 43 a + 20 + \left(15 a^{3} + 55 a^{2} + 58 a + 18\right)\cdot 61 + \left(3 a^{3} + 53 a^{2} + 42 a + 20\right)\cdot 61^{2} + \left(25 a^{3} + 60 a^{2} + 5 a + 37\right)\cdot 61^{3} + \left(29 a^{3} + 54 a^{2} + 44 a + 5\right)\cdot 61^{4} + \left(9 a^{3} + 49 a^{2} + 5 a + 19\right)\cdot 61^{5} + \left(21 a^{3} + 23 a^{2} + 12 a + 58\right)\cdot 61^{6} + \left(24 a^{3} + 19 a^{2} + 22 a + 43\right)\cdot 61^{7} + \left(57 a^{3} + 55 a^{2} + 17 a + 11\right)\cdot 61^{8} + \left(33 a^{3} + 54 a^{2} + 5 a + 9\right)\cdot 61^{9} +O(61^{10})\) 36a^3 + 26a^2 + 43a + 20 + (15a^3 + 55a^2 + 58a + 18)61 + (3a^3 + 53a^2 + 42a + 20)61^2 + (25a^3 + 60a^2 + 5a + 37)61^3 + (29a^3 + 54a^2 + 44a + 5)61^4 + (9a^3 + 49a^2 + 5a + 19)61^5 + (21a^3 + 23a^2 + 12a + 58)61^6 + (24a^3 + 19a^2 + 22a + 43)61^7 + (57a^3 + 55a^2 + 17a + 11)61^8 + (33a^3 + 54a^2 + 5a + 9)*61^9+O(61^10)
$r_{ 10 }$	$=$	\( 10 a^{3} + 42 a^{2} + 39 a + 59 + \left(23 a^{3} + 10 a^{2} + 13 a + 53\right)\cdot 61 + \left(19 a^{3} + 23 a^{2} + 31 a + 11\right)\cdot 61^{2} + \left(17 a^{3} + 27 a^{2} + 34 a + 24\right)\cdot 61^{3} + \left(6 a^{3} + 35 a^{2} + 42 a + 11\right)\cdot 61^{4} + \left(30 a^{3} + a^{2} + 2 a + 56\right)\cdot 61^{5} + \left(4 a^{3} + 40 a^{2} + 8 a + 13\right)\cdot 61^{6} + \left(35 a^{3} + 20 a^{2} + 52 a + 60\right)\cdot 61^{7} + \left(56 a^{3} + 12 a^{2} + 31\right)\cdot 61^{8} + \left(37 a^{3} + 19 a + 33\right)\cdot 61^{9} +O(61^{10})\) 10a^3 + 42a^2 + 39a + 59 + (23a^3 + 10a^2 + 13a + 53)61 + (19a^3 + 23a^2 + 31a + 11)61^2 + (17a^3 + 27a^2 + 34a + 24)61^3 + (6a^3 + 35a^2 + 42a + 11)61^4 + (30a^3 + a^2 + 2a + 56)61^5 + (4a^3 + 40a^2 + 8a + 13)61^6 + (35a^3 + 20a^2 + 52a + 60)61^7 + (56a^3 + 12a^2 + 31)61^8 + (37a^3 + 19a + 33)61^9+O(61^10)