LMFDB - Artin representation 1.231.10t1.b.a

Basic invariants

Dimension:	$1$
Group:	$C_{10}$
Conductor:	$231$$\medspace = 3 \cdot 7 \cdot 11 $
Artin field:	Galois closure of 10.0.9630096522760791.1
Galois orbit size:	$4$
Smallest permutation container:	$C_{10}$
Parity:	odd
Dirichlet character:	$\chi_{231}(62,\cdot)$
Projective image:	$C_1$
Projective field:	Galois closure of $\Q$

Defining polynomial

$f(x)$

$=$

$ x^{10} - x^{9} + 56 x^{8} - 56 x^{7} + 1156 x^{6} - 1156 x^{5} + 10781 x^{4} - 10781 x^{3} + \cdots + 79531 $

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 7.

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

Roots:

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

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	1.231.10t1.b.a
Dimension	$1$
Group	$C_{10}$
Conductor	$231$
Root number	not computed
Indicator	$0$

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