LMFDB - Galois orbit of Artin representations 2.100.10t6.b

Basic invariants

Dimension:	$2$
Group:	$D_5\times C_5$
Conductor:	$100$$\medspace = 2^{2} \cdot 5^{2} $
Artin number field:	Galois closure of 10.0.400000000.1
Galois orbit size:	$4$
Smallest permutation container:	$D_5\times C_5$
Parity:	odd
Projective image:	$D_5$
Projective field:	Galois closure of 5.1.6250000.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 8.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $ x^{5} + 4x + 11 $

Roots:

$r_{ 1 }$	$=$	$ 8 a^{4} + 7 a^{3} + 10 a^{2} + 4 a + 1 + \left(a^{4} + 10 a^{3} + 10 a^{2} + 10 a + 1\right)\cdot 13 + \left(4 a^{4} + 3 a^{3} + 2 a^{2} + 7 a + 3\right)\cdot 13^{2} + \left(a^{4} + 7 a^{3} + 10 a^{2} + 12 a + 4\right)\cdot 13^{3} + \left(5 a^{3} + 2 a + 1\right)\cdot 13^{4} + \left(9 a^{3} + 5 a^{2} + 3 a + 1\right)\cdot 13^{5} + \left(5 a^{4} + a^{3} + 8 a^{2} + 11\right)\cdot 13^{6} + \left(11 a^{4} + 3 a^{3} + 3 a^{2} + 5 a + 7\right)\cdot 13^{7} +O(13^{8})$ 8a^4 + 7a^3 + 10a^2 + 4a + 1 + (a^4 + 10a^3 + 10a^2 + 10a + 1)13 + (4a^4 + 3a^3 + 2a^2 + 7a + 3)13^2 + (a^4 + 7a^3 + 10a^2 + 12a + 4)13^3 + (5a^3 + 2a + 1)13^4 + (9a^3 + 5a^2 + 3a + 1)13^5 + (5a^4 + a^3 + 8a^2 + 11)13^6 + (11a^4 + 3a^3 + 3a^2 + 5a + 7)13^7+O(13^8)
$r_{ 2 }$	$=$	$ 4 a^{4} + 7 a^{3} + 7 a^{2} + 8 a + 7 + \left(2 a^{4} + 9 a^{3} + 6 a^{2} + 11 a + 11\right)\cdot 13 + \left(11 a^{4} + 9 a^{3} + 12 a^{2} + 9 a + 6\right)\cdot 13^{2} + \left(12 a^{4} + 4 a^{3} + 3 a^{2} + 5 a + 12\right)\cdot 13^{3} + \left(7 a^{4} + a^{3} + 5 a^{2} + 8 a + 8\right)\cdot 13^{4} + \left(9 a^{4} + 9 a^{3} + 11 a^{2} + 4 a + 3\right)\cdot 13^{5} + \left(7 a^{4} + 2 a^{3} + 3 a^{2} + 3 a + 6\right)\cdot 13^{6} + \left(12 a^{4} + 11 a^{3} + 11 a^{2} + 11 a + 6\right)\cdot 13^{7} +O(13^{8})$ 4a^4 + 7a^3 + 7a^2 + 8a + 7 + (2a^4 + 9a^3 + 6a^2 + 11a + 11)13 + (11a^4 + 9a^3 + 12a^2 + 9a + 6)13^2 + (12a^4 + 4a^3 + 3a^2 + 5a + 12)13^3 + (7a^4 + a^3 + 5a^2 + 8a + 8)13^4 + (9a^4 + 9a^3 + 11a^2 + 4a + 3)13^5 + (7a^4 + 2a^3 + 3a^2 + 3a + 6)13^6 + (12a^4 + 11a^3 + 11a^2 + 11a + 6)13^7+O(13^8)
$r_{ 3 }$	$=$	$ 6 a^{4} + 9 a^{3} + 10 a + 3 + \left(11 a^{4} + 2 a^{3} + 10 a^{2} + 3 a + 12\right)\cdot 13 + \left(11 a^{4} + 5 a^{3} + 6 a^{2} + 7 a + 3\right)\cdot 13^{2} + \left(6 a^{4} + 2 a^{3} + 12 a^{2} + 8 a + 1\right)\cdot 13^{3} + \left(10 a^{4} + 9 a^{3} + 11 a^{2} + 12 a + 4\right)\cdot 13^{4} + \left(10 a^{4} + 6 a^{3} + 10 a^{2} + 4 a + 2\right)\cdot 13^{5} + \left(6 a^{4} + 11 a^{3} + a^{2} + a + 11\right)\cdot 13^{6} + \left(a^{4} + 6 a^{3} + 3 a^{2} + 7 a + 4\right)\cdot 13^{7} +O(13^{8})$ 6a^4 + 9a^3 + 10a + 3 + (11a^4 + 2a^3 + 10a^2 + 3a + 12)13 + (11a^4 + 5a^3 + 6a^2 + 7a + 3)13^2 + (6a^4 + 2a^3 + 12a^2 + 8a + 1)13^3 + (10a^4 + 9a^3 + 11a^2 + 12a + 4)13^4 + (10a^4 + 6a^3 + 10a^2 + 4a + 2)13^5 + (6a^4 + 11a^3 + a^2 + a + 11)13^6 + (a^4 + 6a^3 + 3a^2 + 7a + 4)*13^7+O(13^8)
$r_{ 4 }$	$=$	$ 6 a^{4} + a^{3} + a^{2} + 7 a + 5 + \left(8 a^{2} + 8 a + 10\right)\cdot 13 + \left(12 a^{4} + 9 a^{3} + 5 a^{2} + 11 a + 12\right)\cdot 13^{2} + \left(11 a^{3} + 2 a^{2} + 8 a + 2\right)\cdot 13^{3} + \left(7 a^{4} + 7 a^{3} + 8 a^{2} + 8 a\right)\cdot 13^{4} + \left(5 a^{4} + 11 a^{3} + 6 a^{2} + 11\right)\cdot 13^{5} + \left(a^{4} + 6 a^{3} + 7 a^{2} + 6 a + 4\right)\cdot 13^{6} + \left(12 a^{4} + 12 a^{3} + 4 a^{2} + 9 a + 2\right)\cdot 13^{7} +O(13^{8})$ 6a^4 + a^3 + a^2 + 7a + 5 + (8a^2 + 8a + 10)13 + (12a^4 + 9a^3 + 5a^2 + 11a + 12)13^2 + (11a^3 + 2a^2 + 8a + 2)13^3 + (7a^4 + 7a^3 + 8a^2 + 8a)13^4 + (5a^4 + 11a^3 + 6a^2 + 11)13^5 + (a^4 + 6a^3 + 7a^2 + 6a + 4)13^6 + (12a^4 + 12a^3 + 4a^2 + 9a + 2)13^7+O(13^8)
$r_{ 5 }$	$=$	$ 3 a^{4} + 5 a^{3} + 10 a + 11 + \left(9 a^{4} + 9 a^{3} + 11 a^{2} + 12 a + 9\right)\cdot 13 + \left(5 a^{3} + 4 a^{2} + 12 a + 7\right)\cdot 13^{2} + \left(9 a^{4} + 6 a^{3} + 7 a^{2} + 5\right)\cdot 13^{3} + \left(9 a^{4} + 10 a^{3} + 7 a^{2} + a + 3\right)\cdot 13^{4} + \left(6 a^{4} + 4 a^{3} + 5 a^{2} + 2 a + 12\right)\cdot 13^{5} + \left(8 a^{4} + 8 a^{3} + 2 a^{2} + a + 11\right)\cdot 13^{6} + \left(6 a^{4} + 12 a^{2} + 5 a + 10\right)\cdot 13^{7} +O(13^{8})$ 3a^4 + 5a^3 + 10a + 11 + (9a^4 + 9a^3 + 11a^2 + 12a + 9)13 + (5a^3 + 4a^2 + 12a + 7)13^2 + (9a^4 + 6a^3 + 7a^2 + 5)13^3 + (9a^4 + 10a^3 + 7a^2 + a + 3)13^4 + (6a^4 + 4a^3 + 5a^2 + 2a + 12)13^5 + (8a^4 + 8a^3 + 2a^2 + a + 11)13^6 + (6a^4 + 12a^2 + 5a + 10)*13^7+O(13^8)
$r_{ 6 }$	$=$	$ 11 a^{4} + 11 a^{3} + 2 a^{2} + 6 a + 8 + \left(8 a^{4} + 9 a^{3} + a^{2} + 8 a + 8\right)\cdot 13 + \left(4 a^{4} + 3 a^{3} + 11 a^{2} + 12\right)\cdot 13^{2} + \left(12 a^{4} + a^{3} + 8 a^{2} + 10 a + 10\right)\cdot 13^{3} + \left(3 a^{4} + 3 a^{3} + 3 a^{2} + 9 a + 5\right)\cdot 13^{4} + \left(7 a^{4} + 5 a^{3} + 6 a^{2} + 4 a + 11\right)\cdot 13^{5} + \left(2 a^{4} + 11 a^{3} + 6 a^{2} + 7 a + 5\right)\cdot 13^{6} + \left(a^{4} + 3 a^{3} + 7 a^{2} + 6 a + 6\right)\cdot 13^{7} +O(13^{8})$ 11a^4 + 11a^3 + 2a^2 + 6a + 8 + (8a^4 + 9a^3 + a^2 + 8a + 8)13 + (4a^4 + 3a^3 + 11a^2 + 12)13^2 + (12a^4 + a^3 + 8a^2 + 10a + 10)13^3 + (3a^4 + 3a^3 + 3a^2 + 9a + 5)13^4 + (7a^4 + 5a^3 + 6a^2 + 4a + 11)13^5 + (2a^4 + 11a^3 + 6a^2 + 7a + 5)13^6 + (a^4 + 3a^3 + 7a^2 + 6a + 6)*13^7+O(13^8)
$r_{ 7 }$	$=$	$ 11 a^{4} + 2 a^{3} + 12 a + 8 + \left(5 a^{4} + 9 a^{3} + 8 a^{2} + 11 a + 1\right)\cdot 13 + \left(4 a^{4} + 3 a^{3} + a^{2} + 5 a + 4\right)\cdot 13^{2} + \left(2 a^{4} + 12 a^{3} + 10 a^{2} + 6 a + 2\right)\cdot 13^{3} + \left(5 a^{4} + 11 a^{3} + 5 a^{2} + 3 a + 7\right)\cdot 13^{4} + \left(6 a^{4} + 7 a^{3} + 2 a^{2} + 2 a + 8\right)\cdot 13^{5} + \left(8 a^{4} + 10 a^{3} + a^{2} + 11 a + 6\right)\cdot 13^{6} + \left(7 a^{4} + 5 a^{3} + 11 a^{2} + 12 a + 11\right)\cdot 13^{7} +O(13^{8})$ 11a^4 + 2a^3 + 12a + 8 + (5a^4 + 9a^3 + 8a^2 + 11a + 1)13 + (4a^4 + 3a^3 + a^2 + 5a + 4)13^2 + (2a^4 + 12a^3 + 10a^2 + 6a + 2)13^3 + (5a^4 + 11a^3 + 5a^2 + 3a + 7)13^4 + (6a^4 + 7a^3 + 2a^2 + 2a + 8)13^5 + (8a^4 + 10a^3 + a^2 + 11a + 6)13^6 + (7a^4 + 5a^3 + 11a^2 + 12a + 11)13^7+O(13^8)
$r_{ 8 }$	$=$	$ 3 a^{4} + a^{3} + 7 a^{2} + 11 a + 9 + \left(3 a^{4} + 12 a^{3} + 5 a^{2} + 8 a + 6\right)\cdot 13 + \left(9 a^{4} + 10 a^{3} + 3 a^{2} + 8\right)\cdot 13^{2} + \left(3 a^{3} + 3 a + 12\right)\cdot 13^{3} + \left(12 a^{4} + 5 a^{2} + a + 8\right)\cdot 13^{4} + \left(a^{4} + 12 a^{3} + 7\right)\cdot 13^{5} + \left(2 a^{4} + 2 a^{3} + 9 a^{2} + 12 a + 6\right)\cdot 13^{6} + \left(8 a^{4} + 5 a^{3} + 12 a^{2} + 10 a + 10\right)\cdot 13^{7} +O(13^{8})$ 3a^4 + a^3 + 7a^2 + 11a + 9 + (3a^4 + 12a^3 + 5a^2 + 8a + 6)13 + (9a^4 + 10a^3 + 3a^2 + 8)13^2 + (3a^3 + 3a + 12)13^3 + (12a^4 + 5a^2 + a + 8)13^4 + (a^4 + 12a^3 + 7)13^5 + (2a^4 + 2a^3 + 9a^2 + 12a + 6)13^6 + (8a^4 + 5a^3 + 12a^2 + 10a + 10)13^7+O(13^8)
$r_{ 9 }$	$=$	$ 3 a^{3} + a + 2 + \left(3 a^{4} + 5 a^{3} + 7 a^{2} + 3 a + 11\right)\cdot 13 + \left(7 a^{4} + 11 a^{3} + 9 a^{2} + 2 a + 9\right)\cdot 13^{2} + \left(6 a^{4} + 2 a^{3} + a^{2} + 2\right)\cdot 13^{3} + \left(10 a^{4} + 3 a^{2} + 12 a + 9\right)\cdot 13^{4} + \left(3 a^{4} + 2 a^{3} + 10 a^{2} + 3 a\right)\cdot 13^{5} + \left(4 a^{4} + 9 a^{3} + 10 a^{2} + 7 a + 3\right)\cdot 13^{6} + \left(12 a^{3} + 3 a^{2} + 7 a + 6\right)\cdot 13^{7} +O(13^{8})$ 3a^3 + a + 2 + (3a^4 + 5a^3 + 7a^2 + 3a + 11)13 + (7a^4 + 11a^3 + 9a^2 + 2a + 9)13^2 + (6a^4 + 2a^3 + a^2 + 2)13^3 + (10a^4 + 3a^2 + 12a + 9)13^4 + (3a^4 + 2a^3 + 10a^2 + 3a)13^5 + (4a^4 + 9a^3 + 10a^2 + 7a + 3)13^6 + (12a^3 + 3a^2 + 7a + 6)13^7+O(13^8)
$r_{ 10 }$	$=$	$ 6 a^{3} + 12 a^{2} + 9 a + 2 + \left(6 a^{4} + 9 a^{3} + 9 a^{2} + 11 a + 5\right)\cdot 13 + \left(12 a^{4} + a^{3} + 6 a^{2} + 5 a + 8\right)\cdot 13^{2} + \left(11 a^{4} + 12 a^{3} + 7 a^{2} + 8 a + 9\right)\cdot 13^{3} + \left(10 a^{4} + a^{3} + 4 a + 2\right)\cdot 13^{4} + \left(12 a^{4} + 9 a^{3} + 6 a^{2} + 12 a + 6\right)\cdot 13^{5} + \left(4 a^{4} + 12 a^{3} + a + 10\right)\cdot 13^{6} + \left(3 a^{4} + 2 a^{3} + 8 a^{2} + 2 a + 10\right)\cdot 13^{7} +O(13^{8})$ 6a^3 + 12a^2 + 9a + 2 + (6a^4 + 9a^3 + 9a^2 + 11a + 5)13 + (12a^4 + a^3 + 6a^2 + 5a + 8)13^2 + (11a^4 + 12a^3 + 7a^2 + 8a + 9)13^3 + (10a^4 + a^3 + 4a + 2)13^4 + (12a^4 + 9a^3 + 6a^2 + 12a + 6)13^5 + (4a^4 + 12a^3 + a + 10)13^6 + (3a^4 + 2a^3 + 8a^2 + 2a + 10)*13^7+O(13^8)

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

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

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 10 }$	Character values
			$c1$	$c2$	$c3$	$c4$
$1$	$1$	$()$	$2$	$2$	$2$	$2$
$5$	$2$	$(1,2)(3,6)(4,10)(5,9)(7,8)$	$0$	$0$	$0$	$0$
$1$	$5$	$(1,7,6,4,5)(2,8,3,10,9)$	$-2 \zeta_{5}^{3} - 2 \zeta_{5}^{2} - 2 \zeta_{5} - 2$	$2 \zeta_{5}^{2}$	$2 \zeta_{5}^{3}$	$2 \zeta_{5}$
$1$	$5$	$(1,6,5,7,4)(2,3,9,8,10)$	$2 \zeta_{5}^{3}$	$-2 \zeta_{5}^{3} - 2 \zeta_{5}^{2} - 2 \zeta_{5} - 2$	$2 \zeta_{5}$	$2 \zeta_{5}^{2}$
$1$	$5$	$(1,4,7,5,6)(2,10,8,9,3)$	$2 \zeta_{5}^{2}$	$2 \zeta_{5}$	$-2 \zeta_{5}^{3} - 2 \zeta_{5}^{2} - 2 \zeta_{5} - 2$	$2 \zeta_{5}^{3}$
$1$	$5$	$(1,5,4,6,7)(2,9,10,3,8)$	$2 \zeta_{5}$	$2 \zeta_{5}^{3}$	$2 \zeta_{5}^{2}$	$-2 \zeta_{5}^{3} - 2 \zeta_{5}^{2} - 2 \zeta_{5} - 2$
$2$	$5$	$(1,7,6,4,5)(2,9,10,3,8)$	$-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$	$\zeta_{5}^{3} + \zeta_{5}^{2}$	$\zeta_{5}^{3} + \zeta_{5}^{2}$	$-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$
$2$	$5$	$(1,6,5,7,4)(2,10,8,9,3)$	$\zeta_{5}^{3} + \zeta_{5}^{2}$	$-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$	$-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$	$\zeta_{5}^{3} + \zeta_{5}^{2}$
$2$	$5$	$(2,8,3,10,9)$	$-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5}$	$\zeta_{5}^{2} + 1$	$\zeta_{5}^{3} + 1$	$\zeta_{5} + 1$
$2$	$5$	$(2,3,9,8,10)$	$\zeta_{5}^{3} + 1$	$-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5}$	$\zeta_{5} + 1$	$\zeta_{5}^{2} + 1$
$2$	$5$	$(2,10,8,9,3)$	$\zeta_{5}^{2} + 1$	$\zeta_{5} + 1$	$-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5}$	$\zeta_{5}^{3} + 1$
$2$	$5$	$(2,9,10,3,8)$	$\zeta_{5} + 1$	$\zeta_{5}^{3} + 1$	$\zeta_{5}^{2} + 1$	$-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5}$
$2$	$5$	$(1,5,4,6,7)(2,3,9,8,10)$	$\zeta_{5}^{3} + \zeta_{5}$	$-\zeta_{5}^{2} - \zeta_{5} - 1$	$\zeta_{5}^{2} + \zeta_{5}$	$-\zeta_{5}^{3} - \zeta_{5} - 1$
$2$	$5$	$(1,4,7,5,6)(2,9,10,3,8)$	$\zeta_{5}^{2} + \zeta_{5}$	$\zeta_{5}^{3} + \zeta_{5}$	$-\zeta_{5}^{3} - \zeta_{5} - 1$	$-\zeta_{5}^{2} - \zeta_{5} - 1$
$2$	$5$	$(1,6,5,7,4)(2,8,3,10,9)$	$-\zeta_{5}^{2} - \zeta_{5} - 1$	$-\zeta_{5}^{3} - \zeta_{5} - 1$	$\zeta_{5}^{3} + \zeta_{5}$	$\zeta_{5}^{2} + \zeta_{5}$
$2$	$5$	$(1,7,6,4,5)(2,10,8,9,3)$	$-\zeta_{5}^{3} - \zeta_{5} - 1$	$\zeta_{5}^{2} + \zeta_{5}$	$-\zeta_{5}^{2} - \zeta_{5} - 1$	$\zeta_{5}^{3} + \zeta_{5}$
$5$	$10$	$(1,8,7,3,6,10,4,9,5,2)$	$0$	$0$	$0$	$0$
$5$	$10$	$(1,3,4,2,7,10,5,8,6,9)$	$0$	$0$	$0$	$0$
$5$	$10$	$(1,9,6,8,5,10,7,2,4,3)$	$0$	$0$	$0$	$0$
$5$	$10$	$(1,2,5,9,4,10,6,3,7,8)$	$0$	$0$	$0$	$0$

The blue line marks the conjugacy class containing complex conjugation.

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Basic invariants

Galois action

Roots of 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	2.100.10t6.b
Dimension	$2$
Group	$D_5\times C_5$
Conductor	$100$
Indicator	$0$

$r_{ 1 }$	$=$	\( 8 a^{4} + 7 a^{3} + 10 a^{2} + 4 a + 1 + \left(a^{4} + 10 a^{3} + 10 a^{2} + 10 a + 1\right)\cdot 13 + \left(4 a^{4} + 3 a^{3} + 2 a^{2} + 7 a + 3\right)\cdot 13^{2} + \left(a^{4} + 7 a^{3} + 10 a^{2} + 12 a + 4\right)\cdot 13^{3} + \left(5 a^{3} + 2 a + 1\right)\cdot 13^{4} + \left(9 a^{3} + 5 a^{2} + 3 a + 1\right)\cdot 13^{5} + \left(5 a^{4} + a^{3} + 8 a^{2} + 11\right)\cdot 13^{6} + \left(11 a^{4} + 3 a^{3} + 3 a^{2} + 5 a + 7\right)\cdot 13^{7} +O(13^{8})\) 8a^4 + 7a^3 + 10a^2 + 4a + 1 + (a^4 + 10a^3 + 10a^2 + 10a + 1)13 + (4a^4 + 3a^3 + 2a^2 + 7a + 3)13^2 + (a^4 + 7a^3 + 10a^2 + 12a + 4)13^3 + (5a^3 + 2a + 1)13^4 + (9a^3 + 5a^2 + 3a + 1)13^5 + (5a^4 + a^3 + 8a^2 + 11)13^6 + (11a^4 + 3a^3 + 3a^2 + 5a + 7)13^7+O(13^8)
$r_{ 2 }$	$=$	\( 4 a^{4} + 7 a^{3} + 7 a^{2} + 8 a + 7 + \left(2 a^{4} + 9 a^{3} + 6 a^{2} + 11 a + 11\right)\cdot 13 + \left(11 a^{4} + 9 a^{3} + 12 a^{2} + 9 a + 6\right)\cdot 13^{2} + \left(12 a^{4} + 4 a^{3} + 3 a^{2} + 5 a + 12\right)\cdot 13^{3} + \left(7 a^{4} + a^{3} + 5 a^{2} + 8 a + 8\right)\cdot 13^{4} + \left(9 a^{4} + 9 a^{3} + 11 a^{2} + 4 a + 3\right)\cdot 13^{5} + \left(7 a^{4} + 2 a^{3} + 3 a^{2} + 3 a + 6\right)\cdot 13^{6} + \left(12 a^{4} + 11 a^{3} + 11 a^{2} + 11 a + 6\right)\cdot 13^{7} +O(13^{8})\) 4a^4 + 7a^3 + 7a^2 + 8a + 7 + (2a^4 + 9a^3 + 6a^2 + 11a + 11)13 + (11a^4 + 9a^3 + 12a^2 + 9a + 6)13^2 + (12a^4 + 4a^3 + 3a^2 + 5a + 12)13^3 + (7a^4 + a^3 + 5a^2 + 8a + 8)13^4 + (9a^4 + 9a^3 + 11a^2 + 4a + 3)13^5 + (7a^4 + 2a^3 + 3a^2 + 3a + 6)13^6 + (12a^4 + 11a^3 + 11a^2 + 11a + 6)13^7+O(13^8)
$r_{ 3 }$	$=$	\( 6 a^{4} + 9 a^{3} + 10 a + 3 + \left(11 a^{4} + 2 a^{3} + 10 a^{2} + 3 a + 12\right)\cdot 13 + \left(11 a^{4} + 5 a^{3} + 6 a^{2} + 7 a + 3\right)\cdot 13^{2} + \left(6 a^{4} + 2 a^{3} + 12 a^{2} + 8 a + 1\right)\cdot 13^{3} + \left(10 a^{4} + 9 a^{3} + 11 a^{2} + 12 a + 4\right)\cdot 13^{4} + \left(10 a^{4} + 6 a^{3} + 10 a^{2} + 4 a + 2\right)\cdot 13^{5} + \left(6 a^{4} + 11 a^{3} + a^{2} + a + 11\right)\cdot 13^{6} + \left(a^{4} + 6 a^{3} + 3 a^{2} + 7 a + 4\right)\cdot 13^{7} +O(13^{8})\) 6a^4 + 9a^3 + 10a + 3 + (11a^4 + 2a^3 + 10a^2 + 3a + 12)13 + (11a^4 + 5a^3 + 6a^2 + 7a + 3)13^2 + (6a^4 + 2a^3 + 12a^2 + 8a + 1)13^3 + (10a^4 + 9a^3 + 11a^2 + 12a + 4)13^4 + (10a^4 + 6a^3 + 10a^2 + 4a + 2)13^5 + (6a^4 + 11a^3 + a^2 + a + 11)13^6 + (a^4 + 6a^3 + 3a^2 + 7a + 4)*13^7+O(13^8)
$r_{ 4 }$	$=$	\( 6 a^{4} + a^{3} + a^{2} + 7 a + 5 + \left(8 a^{2} + 8 a + 10\right)\cdot 13 + \left(12 a^{4} + 9 a^{3} + 5 a^{2} + 11 a + 12\right)\cdot 13^{2} + \left(11 a^{3} + 2 a^{2} + 8 a + 2\right)\cdot 13^{3} + \left(7 a^{4} + 7 a^{3} + 8 a^{2} + 8 a\right)\cdot 13^{4} + \left(5 a^{4} + 11 a^{3} + 6 a^{2} + 11\right)\cdot 13^{5} + \left(a^{4} + 6 a^{3} + 7 a^{2} + 6 a + 4\right)\cdot 13^{6} + \left(12 a^{4} + 12 a^{3} + 4 a^{2} + 9 a + 2\right)\cdot 13^{7} +O(13^{8})\) 6a^4 + a^3 + a^2 + 7a + 5 + (8a^2 + 8a + 10)13 + (12a^4 + 9a^3 + 5a^2 + 11a + 12)13^2 + (11a^3 + 2a^2 + 8a + 2)13^3 + (7a^4 + 7a^3 + 8a^2 + 8a)13^4 + (5a^4 + 11a^3 + 6a^2 + 11)13^5 + (a^4 + 6a^3 + 7a^2 + 6a + 4)13^6 + (12a^4 + 12a^3 + 4a^2 + 9a + 2)13^7+O(13^8)
$r_{ 5 }$	$=$	\( 3 a^{4} + 5 a^{3} + 10 a + 11 + \left(9 a^{4} + 9 a^{3} + 11 a^{2} + 12 a + 9\right)\cdot 13 + \left(5 a^{3} + 4 a^{2} + 12 a + 7\right)\cdot 13^{2} + \left(9 a^{4} + 6 a^{3} + 7 a^{2} + 5\right)\cdot 13^{3} + \left(9 a^{4} + 10 a^{3} + 7 a^{2} + a + 3\right)\cdot 13^{4} + \left(6 a^{4} + 4 a^{3} + 5 a^{2} + 2 a + 12\right)\cdot 13^{5} + \left(8 a^{4} + 8 a^{3} + 2 a^{2} + a + 11\right)\cdot 13^{6} + \left(6 a^{4} + 12 a^{2} + 5 a + 10\right)\cdot 13^{7} +O(13^{8})\) 3a^4 + 5a^3 + 10a + 11 + (9a^4 + 9a^3 + 11a^2 + 12a + 9)13 + (5a^3 + 4a^2 + 12a + 7)13^2 + (9a^4 + 6a^3 + 7a^2 + 5)13^3 + (9a^4 + 10a^3 + 7a^2 + a + 3)13^4 + (6a^4 + 4a^3 + 5a^2 + 2a + 12)13^5 + (8a^4 + 8a^3 + 2a^2 + a + 11)13^6 + (6a^4 + 12a^2 + 5a + 10)*13^7+O(13^8)
$r_{ 6 }$	$=$	\( 11 a^{4} + 11 a^{3} + 2 a^{2} + 6 a + 8 + \left(8 a^{4} + 9 a^{3} + a^{2} + 8 a + 8\right)\cdot 13 + \left(4 a^{4} + 3 a^{3} + 11 a^{2} + 12\right)\cdot 13^{2} + \left(12 a^{4} + a^{3} + 8 a^{2} + 10 a + 10\right)\cdot 13^{3} + \left(3 a^{4} + 3 a^{3} + 3 a^{2} + 9 a + 5\right)\cdot 13^{4} + \left(7 a^{4} + 5 a^{3} + 6 a^{2} + 4 a + 11\right)\cdot 13^{5} + \left(2 a^{4} + 11 a^{3} + 6 a^{2} + 7 a + 5\right)\cdot 13^{6} + \left(a^{4} + 3 a^{3} + 7 a^{2} + 6 a + 6\right)\cdot 13^{7} +O(13^{8})\) 11a^4 + 11a^3 + 2a^2 + 6a + 8 + (8a^4 + 9a^3 + a^2 + 8a + 8)13 + (4a^4 + 3a^3 + 11a^2 + 12)13^2 + (12a^4 + a^3 + 8a^2 + 10a + 10)13^3 + (3a^4 + 3a^3 + 3a^2 + 9a + 5)13^4 + (7a^4 + 5a^3 + 6a^2 + 4a + 11)13^5 + (2a^4 + 11a^3 + 6a^2 + 7a + 5)13^6 + (a^4 + 3a^3 + 7a^2 + 6a + 6)*13^7+O(13^8)
$r_{ 7 }$	$=$	\( 11 a^{4} + 2 a^{3} + 12 a + 8 + \left(5 a^{4} + 9 a^{3} + 8 a^{2} + 11 a + 1\right)\cdot 13 + \left(4 a^{4} + 3 a^{3} + a^{2} + 5 a + 4\right)\cdot 13^{2} + \left(2 a^{4} + 12 a^{3} + 10 a^{2} + 6 a + 2\right)\cdot 13^{3} + \left(5 a^{4} + 11 a^{3} + 5 a^{2} + 3 a + 7\right)\cdot 13^{4} + \left(6 a^{4} + 7 a^{3} + 2 a^{2} + 2 a + 8\right)\cdot 13^{5} + \left(8 a^{4} + 10 a^{3} + a^{2} + 11 a + 6\right)\cdot 13^{6} + \left(7 a^{4} + 5 a^{3} + 11 a^{2} + 12 a + 11\right)\cdot 13^{7} +O(13^{8})\) 11a^4 + 2a^3 + 12a + 8 + (5a^4 + 9a^3 + 8a^2 + 11a + 1)13 + (4a^4 + 3a^3 + a^2 + 5a + 4)13^2 + (2a^4 + 12a^3 + 10a^2 + 6a + 2)13^3 + (5a^4 + 11a^3 + 5a^2 + 3a + 7)13^4 + (6a^4 + 7a^3 + 2a^2 + 2a + 8)13^5 + (8a^4 + 10a^3 + a^2 + 11a + 6)13^6 + (7a^4 + 5a^3 + 11a^2 + 12a + 11)13^7+O(13^8)
$r_{ 8 }$	$=$	\( 3 a^{4} + a^{3} + 7 a^{2} + 11 a + 9 + \left(3 a^{4} + 12 a^{3} + 5 a^{2} + 8 a + 6\right)\cdot 13 + \left(9 a^{4} + 10 a^{3} + 3 a^{2} + 8\right)\cdot 13^{2} + \left(3 a^{3} + 3 a + 12\right)\cdot 13^{3} + \left(12 a^{4} + 5 a^{2} + a + 8\right)\cdot 13^{4} + \left(a^{4} + 12 a^{3} + 7\right)\cdot 13^{5} + \left(2 a^{4} + 2 a^{3} + 9 a^{2} + 12 a + 6\right)\cdot 13^{6} + \left(8 a^{4} + 5 a^{3} + 12 a^{2} + 10 a + 10\right)\cdot 13^{7} +O(13^{8})\) 3a^4 + a^3 + 7a^2 + 11a + 9 + (3a^4 + 12a^3 + 5a^2 + 8a + 6)13 + (9a^4 + 10a^3 + 3a^2 + 8)13^2 + (3a^3 + 3a + 12)13^3 + (12a^4 + 5a^2 + a + 8)13^4 + (a^4 + 12a^3 + 7)13^5 + (2a^4 + 2a^3 + 9a^2 + 12a + 6)13^6 + (8a^4 + 5a^3 + 12a^2 + 10a + 10)13^7+O(13^8)
$r_{ 9 }$	$=$	\( 3 a^{3} + a + 2 + \left(3 a^{4} + 5 a^{3} + 7 a^{2} + 3 a + 11\right)\cdot 13 + \left(7 a^{4} + 11 a^{3} + 9 a^{2} + 2 a + 9\right)\cdot 13^{2} + \left(6 a^{4} + 2 a^{3} + a^{2} + 2\right)\cdot 13^{3} + \left(10 a^{4} + 3 a^{2} + 12 a + 9\right)\cdot 13^{4} + \left(3 a^{4} + 2 a^{3} + 10 a^{2} + 3 a\right)\cdot 13^{5} + \left(4 a^{4} + 9 a^{3} + 10 a^{2} + 7 a + 3\right)\cdot 13^{6} + \left(12 a^{3} + 3 a^{2} + 7 a + 6\right)\cdot 13^{7} +O(13^{8})\) 3a^3 + a + 2 + (3a^4 + 5a^3 + 7a^2 + 3a + 11)13 + (7a^4 + 11a^3 + 9a^2 + 2a + 9)13^2 + (6a^4 + 2a^3 + a^2 + 2)13^3 + (10a^4 + 3a^2 + 12a + 9)13^4 + (3a^4 + 2a^3 + 10a^2 + 3a)13^5 + (4a^4 + 9a^3 + 10a^2 + 7a + 3)13^6 + (12a^3 + 3a^2 + 7a + 6)13^7+O(13^8)
$r_{ 10 }$	$=$	\( 6 a^{3} + 12 a^{2} + 9 a + 2 + \left(6 a^{4} + 9 a^{3} + 9 a^{2} + 11 a + 5\right)\cdot 13 + \left(12 a^{4} + a^{3} + 6 a^{2} + 5 a + 8\right)\cdot 13^{2} + \left(11 a^{4} + 12 a^{3} + 7 a^{2} + 8 a + 9\right)\cdot 13^{3} + \left(10 a^{4} + a^{3} + 4 a + 2\right)\cdot 13^{4} + \left(12 a^{4} + 9 a^{3} + 6 a^{2} + 12 a + 6\right)\cdot 13^{5} + \left(4 a^{4} + 12 a^{3} + a + 10\right)\cdot 13^{6} + \left(3 a^{4} + 2 a^{3} + 8 a^{2} + 2 a + 10\right)\cdot 13^{7} +O(13^{8})\) 6a^3 + 12a^2 + 9a + 2 + (6a^4 + 9a^3 + 9a^2 + 11a + 5)13 + (12a^4 + a^3 + 6a^2 + 5a + 8)13^2 + (11a^4 + 12a^3 + 7a^2 + 8a + 9)13^3 + (10a^4 + a^3 + 4a + 2)13^4 + (12a^4 + 9a^3 + 6a^2 + 12a + 6)13^5 + (4a^4 + 12a^3 + a + 10)13^6 + (3a^4 + 2a^3 + 8a^2 + 2a + 10)*13^7+O(13^8)