LMFDB - Galois orbit of Artin representations 2.468.12t18.a

Basic invariants

Dimension:	$2$
Group:	$C_6\times S_3$
Conductor:	$468$$\medspace = 2^{2} \cdot 3^{2} \cdot 13 $
Artin number field:	Galois closure of 12.0.85282689024.1
Galois orbit size:	$2$
Smallest permutation container:	$C_6\times S_3$
Parity:	odd
Projective image:	$S_3$
Projective field:	Galois closure of 3.1.676.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $ x^{6} + 17x^{3} + 17x^{2} + 6x + 2 $

Roots:

$r_{ 1 }$	$=$	$ 7 a^{5} + 7 a^{4} + 18 a^{3} + 11 a^{2} + 2 a + 14 + \left(13 a^{5} + 5 a^{4} + a^{3} + 16 a^{2} + 4 a + 1\right)\cdot 19 + \left(7 a^{5} + 7 a^{4} + 18 a^{3} + a^{2} + 5 a + 12\right)\cdot 19^{2} + \left(14 a^{5} + 12 a^{4} + 10 a^{2} + 3 a + 14\right)\cdot 19^{3} + \left(14 a^{5} + 11 a^{4} + 7 a^{3} + 7 a^{2} + 10 a + 18\right)\cdot 19^{4} +O(19^{5})$ 7a^5 + 7a^4 + 18a^3 + 11a^2 + 2a + 14 + (13a^5 + 5a^4 + a^3 + 16a^2 + 4a + 1)19 + (7a^5 + 7a^4 + 18a^3 + a^2 + 5a + 12)19^2 + (14a^5 + 12a^4 + 10a^2 + 3a + 14)19^3 + (14a^5 + 11a^4 + 7a^3 + 7a^2 + 10a + 18)19^4+O(19^5)
$r_{ 2 }$	$=$	$ 13 a^{5} + 9 a^{4} + 15 a^{3} + 7 a + \left(11 a^{5} + 9 a^{4} + 6 a^{3} + 6 a^{2} + 5\right)\cdot 19 + \left(13 a^{5} + 5 a^{4} + 9 a^{3} + a + 7\right)\cdot 19^{2} + \left(8 a^{5} + a^{4} + 12 a^{3} + 17 a^{2} + 12 a + 12\right)\cdot 19^{3} + \left(16 a^{5} + 17 a^{4} + 12 a^{3} + 18 a^{2} + a + 9\right)\cdot 19^{4} +O(19^{5})$ 13a^5 + 9a^4 + 15a^3 + 7a + (11a^5 + 9a^4 + 6a^3 + 6a^2 + 5)19 + (13a^5 + 5a^4 + 9a^3 + a + 7)19^2 + (8a^5 + a^4 + 12a^3 + 17a^2 + 12a + 12)19^3 + (16a^5 + 17a^4 + 12a^3 + 18a^2 + a + 9)*19^4+O(19^5)
$r_{ 3 }$	$=$	$ 8 a^{5} + 16 a^{4} + 4 a^{3} + 11 a^{2} + 4 a + 2 + \left(2 a^{5} + 13 a^{4} + 15 a^{3} + a^{2} + 10\right)\cdot 19 + \left(7 a^{5} + 9 a^{4} + 8 a^{3} + 15 a^{2} + 5 a + 8\right)\cdot 19^{2} + \left(4 a^{5} + 10 a^{4} + 7 a^{3} + 15 a^{2} + 2 a + 14\right)\cdot 19^{3} + \left(a^{5} + a^{4} + 6 a^{3} + 14 a^{2} + 6 a + 14\right)\cdot 19^{4} +O(19^{5})$ 8a^5 + 16a^4 + 4a^3 + 11a^2 + 4a + 2 + (2a^5 + 13a^4 + 15a^3 + a^2 + 10)19 + (7a^5 + 9a^4 + 8a^3 + 15a^2 + 5a + 8)19^2 + (4a^5 + 10a^4 + 7a^3 + 15a^2 + 2a + 14)19^3 + (a^5 + a^4 + 6a^3 + 14a^2 + 6a + 14)*19^4+O(19^5)
$r_{ 4 }$	$=$	$ 2 a^{5} + 8 a^{4} + 11 a^{3} + 5 a^{2} + 2 a + 1 + \left(15 a^{5} + 10 a^{4} + 5 a^{3} + 17 a^{2} + 15 a + 13\right)\cdot 19 + \left(9 a^{5} + 2 a^{4} + a^{3} + 18 a^{2} + 14 a + 9\right)\cdot 19^{2} + \left(5 a^{3} + 6 a^{2} + 4 a + 15\right)\cdot 19^{3} + \left(5 a^{5} + 3 a^{4} + 11 a^{3} + 9 a^{2} + 2 a + 18\right)\cdot 19^{4} +O(19^{5})$ 2a^5 + 8a^4 + 11a^3 + 5a^2 + 2a + 1 + (15a^5 + 10a^4 + 5a^3 + 17a^2 + 15a + 13)19 + (9a^5 + 2a^4 + a^3 + 18a^2 + 14a + 9)19^2 + (5a^3 + 6a^2 + 4a + 15)19^3 + (5a^5 + 3a^4 + 11a^3 + 9a^2 + 2a + 18)19^4+O(19^5)
$r_{ 5 }$	$=$	$ 11 a^{5} + 11 a^{4} + 12 a^{3} + a^{2} + 14 a + 3 + \left(8 a^{5} + 9 a^{4} + 17 a^{3} + 15 a^{2} + a + 13\right)\cdot 19 + \left(18 a^{5} + 9 a^{4} + 15 a^{3} + 15 a^{2} + 17 a + 18\right)\cdot 19^{2} + \left(5 a^{5} + 5 a^{4} + 6 a^{3} + 5 a^{2} + 15 a + 10\right)\cdot 19^{3} + \left(2 a^{4} + 18 a^{3} + 6 a + 12\right)\cdot 19^{4} +O(19^{5})$ 11a^5 + 11a^4 + 12a^3 + a^2 + 14a + 3 + (8a^5 + 9a^4 + 17a^3 + 15a^2 + a + 13)19 + (18a^5 + 9a^4 + 15a^3 + 15a^2 + 17a + 18)19^2 + (5a^5 + 5a^4 + 6a^3 + 5a^2 + 15a + 10)19^3 + (2a^4 + 18a^3 + 6a + 12)*19^4+O(19^5)
$r_{ 6 }$	$=$	$ 15 a^{5} + 6 a^{4} + 10 a^{3} + 11 a + \left(10 a^{5} + 11 a^{4} + 6 a^{3} + 4 a^{2} + 12 a + 16\right)\cdot 19 + \left(15 a^{4} + 6 a^{3} + 16 a^{2} + 4 a + 11\right)\cdot 19^{2} + \left(5 a^{5} + 5 a^{4} + 9 a^{3} + 13 a^{2} + 7 a + 2\right)\cdot 19^{3} + \left(12 a^{5} + 15 a^{4} + 14 a^{3} + 9 a^{2} + 16 a\right)\cdot 19^{4} +O(19^{5})$ 15a^5 + 6a^4 + 10a^3 + 11a + (10a^5 + 11a^4 + 6a^3 + 4a^2 + 12a + 16)19 + (15a^4 + 6a^3 + 16a^2 + 4a + 11)19^2 + (5a^5 + 5a^4 + 9a^3 + 13a^2 + 7a + 2)19^3 + (12a^5 + 15a^4 + 14a^3 + 9a^2 + 16a)*19^4+O(19^5)
$r_{ 7 }$	$=$	$ 10 a^{5} + 13 a^{4} + 13 a^{3} + 18 a^{2} + 4 a + 7 + \left(7 a^{5} + 4 a^{4} + 6 a^{3} + 8 a^{2} + 2 a + 11\right)\cdot 19 + \left(12 a^{5} + 5 a^{4} + 12 a^{3} + 4 a^{2} + 10 a + 1\right)\cdot 19^{2} + \left(5 a^{5} + 15 a^{4} + 17 a^{3} + 8 a^{2} + 12 a + 3\right)\cdot 19^{3} + \left(17 a^{5} + 3 a^{4} + 15 a^{3} + 2 a^{2} + 7 a + 2\right)\cdot 19^{4} +O(19^{5})$ 10a^5 + 13a^4 + 13a^3 + 18a^2 + 4a + 7 + (7a^5 + 4a^4 + 6a^3 + 8a^2 + 2a + 11)19 + (12a^5 + 5a^4 + 12a^3 + 4a^2 + 10a + 1)19^2 + (5a^5 + 15a^4 + 17a^3 + 8a^2 + 12a + 3)19^3 + (17a^5 + 3a^4 + 15a^3 + 2a^2 + 7a + 2)*19^4+O(19^5)
$r_{ 8 }$	$=$	$ 13 a^{5} + 15 a^{4} + 13 a^{3} + 16 a^{2} + 17 a + 13 + \left(15 a^{5} + 16 a^{4} + 8 a^{3} + 14 a^{2} + a + 3\right)\cdot 19 + \left(10 a^{5} + 15 a^{4} + 5 a^{2} + 2 a + 13\right)\cdot 19^{2} + \left(15 a^{5} + 12 a^{4} + 15 a^{3} + 17 a^{2} + 5 a + 2\right)\cdot 19^{3} + \left(3 a^{5} + 9 a^{4} + 10 a^{3} + 15 a^{2} + 5 a + 5\right)\cdot 19^{4} +O(19^{5})$ 13a^5 + 15a^4 + 13a^3 + 16a^2 + 17a + 13 + (15a^5 + 16a^4 + 8a^3 + 14a^2 + a + 3)19 + (10a^5 + 15a^4 + 5a^2 + 2a + 13)19^2 + (15a^5 + 12a^4 + 15a^3 + 17a^2 + 5a + 2)19^3 + (3a^5 + 9a^4 + 10a^3 + 15a^2 + 5a + 5)*19^4+O(19^5)
$r_{ 9 }$	$=$	$ 14 a^{5} + 18 a^{4} + a^{3} + 16 a^{2} + 14 a + 7 + \left(2 a^{5} + 2 a^{4} + 14 a^{3} + 5 a^{2} + 2 a + 11\right)\cdot 19 + \left(5 a^{5} + 2 a^{4} + 17 a^{3} + 8 a^{2} + 2 a + 4\right)\cdot 19^{2} + \left(9 a^{5} + 17 a^{4} + 17 a^{3} + 16 a^{2} + 3\right)\cdot 19^{3} + \left(18 a^{5} + 10 a^{4} + 3 a^{3} + 3 a^{2} + 3 a + 15\right)\cdot 19^{4} +O(19^{5})$ 14a^5 + 18a^4 + a^3 + 16a^2 + 14a + 7 + (2a^5 + 2a^4 + 14a^3 + 5a^2 + 2a + 11)19 + (5a^5 + 2a^4 + 17a^3 + 8a^2 + 2a + 4)19^2 + (9a^5 + 17a^4 + 17a^3 + 16a^2 + 3)19^3 + (18a^5 + 10a^4 + 3a^3 + 3a^2 + 3a + 15)*19^4+O(19^5)
$r_{ 10 }$	$=$	$ 12 a^{5} + 5 a^{4} + 6 a^{3} + 7 a^{2} + 6 a + 3 + \left(12 a^{5} + 10 a^{4} + 8 a^{3} + 12 a^{2} + 14 a + 3\right)\cdot 19 + \left(2 a^{5} + a^{4} + 15 a^{3} + 6 a^{2} + 4 a + 13\right)\cdot 19^{2} + \left(12 a^{5} + 15 a^{4} + 5 a^{3} + 16 a^{2} + 8\right)\cdot 19^{3} + \left(18 a^{5} + 11 a^{4} + 18 a^{3} + 2 a^{2} + 11 a + 2\right)\cdot 19^{4} +O(19^{5})$ 12a^5 + 5a^4 + 6a^3 + 7a^2 + 6a + 3 + (12a^5 + 10a^4 + 8a^3 + 12a^2 + 14a + 3)19 + (2a^5 + a^4 + 15a^3 + 6a^2 + 4a + 13)19^2 + (12a^5 + 15a^4 + 5a^3 + 16a^2 + 8)19^3 + (18a^5 + 11a^4 + 18a^3 + 2a^2 + 11a + 2)*19^4+O(19^5)
$r_{ 11 }$	$=$	$ 13 a^{5} + 15 a^{4} + 15 a^{3} + 12 a^{2} + 9 a + 11 + \left(12 a^{5} + 18 a^{4} + 13 a^{3} + 9 a^{2} + 18 a + 13\right)\cdot 19 + \left(3 a^{5} + 7 a^{4} + 11 a^{3} + 6 a^{2} + 17 a + 2\right)\cdot 19^{2} + \left(12 a^{5} + 9 a^{4} + 13 a^{3} + 16 a^{2} + 7 a + 11\right)\cdot 19^{3} + \left(17 a^{5} + a^{4} + 18 a^{2} + 12 a + 3\right)\cdot 19^{4} +O(19^{5})$ 13a^5 + 15a^4 + 15a^3 + 12a^2 + 9a + 11 + (12a^5 + 18a^4 + 13a^3 + 9a^2 + 18a + 13)19 + (3a^5 + 7a^4 + 11a^3 + 6a^2 + 17a + 2)19^2 + (12a^5 + 9a^4 + 13a^3 + 16a^2 + 7a + 11)19^3 + (17a^5 + a^4 + 18a^2 + 12a + 3)*19^4+O(19^5)
$r_{ 12 }$	$=$	$ 15 a^{5} + 10 a^{4} + 15 a^{3} + 17 a^{2} + 5 a + 15 + \left(8 a^{3} + a^{2} + 2 a + 11\right)\cdot 19 + \left(3 a^{5} + 12 a^{4} + 15 a^{3} + 14 a^{2} + 10 a + 10\right)\cdot 19^{2} + \left(a^{5} + 8 a^{4} + a^{3} + 7 a^{2} + 4 a + 14\right)\cdot 19^{3} + \left(7 a^{5} + 6 a^{4} + 13 a^{3} + 9 a^{2} + 12 a + 10\right)\cdot 19^{4} +O(19^{5})$ 15a^5 + 10a^4 + 15a^3 + 17a^2 + 5a + 15 + (8a^3 + a^2 + 2a + 11)19 + (3a^5 + 12a^4 + 15a^3 + 14a^2 + 10a + 10)19^2 + (a^5 + 8a^4 + a^3 + 7a^2 + 4a + 14)19^3 + (7a^5 + 6a^4 + 13a^3 + 9a^2 + 12a + 10)19^4+O(19^5)

Generators of the action on the roots $r_1, \ldots, r_{ 12 }$

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

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 12 }$	Character values
			$c1$	$c2$
$1$	$1$	$()$	$2$	$2$
$1$	$2$	$(1,5)(2,6)(3,10)(4,9)(7,11)(8,12)$	$-2$	$-2$
$3$	$2$	$(1,7)(2,8)(3,9)(4,10)(5,11)(6,12)$	$0$	$0$
$3$	$2$	$(1,3)(2,9)(4,6)(5,10)(7,12)(8,11)$	$0$	$0$
$1$	$3$	$(1,8,4)(2,10,7)(3,11,6)(5,12,9)$	$2 \zeta_{3}$	$-2 \zeta_{3} - 2$
$1$	$3$	$(1,4,8)(2,7,10)(3,6,11)(5,9,12)$	$-2 \zeta_{3} - 2$	$2 \zeta_{3}$
$2$	$3$	$(1,4,8)(5,9,12)$	$-\zeta_{3}$	$\zeta_{3} + 1$
$2$	$3$	$(1,8,4)(5,12,9)$	$\zeta_{3} + 1$	$-\zeta_{3}$
$2$	$3$	$(1,4,8)(2,10,7)(3,11,6)(5,9,12)$	$-1$	$-1$
$1$	$6$	$(1,9,8,5,4,12)(2,11,10,6,7,3)$	$2 \zeta_{3} + 2$	$-2 \zeta_{3}$
$1$	$6$	$(1,12,4,5,8,9)(2,3,7,6,10,11)$	$-2 \zeta_{3}$	$2 \zeta_{3} + 2$
$2$	$6$	$(1,12,4,5,8,9)(2,6)(3,10)(7,11)$	$-\zeta_{3} - 1$	$\zeta_{3}$
$2$	$6$	$(1,9,8,5,4,12)(2,6)(3,10)(7,11)$	$\zeta_{3}$	$-\zeta_{3} - 1$
$2$	$6$	$(1,12,4,5,8,9)(2,11,10,6,7,3)$	$1$	$1$
$3$	$6$	$(1,6,8,3,4,11)(2,12,10,9,7,5)$	$0$	$0$
$3$	$6$	$(1,11,4,3,8,6)(2,5,7,9,10,12)$	$0$	$0$
$3$	$6$	$(1,10,4,2,8,7)(3,9,6,12,11,5)$	$0$	$0$
$3$	$6$	$(1,7,8,2,4,10)(3,5,11,12,6,9)$	$0$	$0$

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 $r_1, \ldots, r_{ 12 }$

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	2.468.12t18.a
Dimension	$2$
Group	$C_6\times S_3$
Conductor	$468$
Indicator	$0$

$r_{ 1 }$	$=$	\( 7 a^{5} + 7 a^{4} + 18 a^{3} + 11 a^{2} + 2 a + 14 + \left(13 a^{5} + 5 a^{4} + a^{3} + 16 a^{2} + 4 a + 1\right)\cdot 19 + \left(7 a^{5} + 7 a^{4} + 18 a^{3} + a^{2} + 5 a + 12\right)\cdot 19^{2} + \left(14 a^{5} + 12 a^{4} + 10 a^{2} + 3 a + 14\right)\cdot 19^{3} + \left(14 a^{5} + 11 a^{4} + 7 a^{3} + 7 a^{2} + 10 a + 18\right)\cdot 19^{4} +O(19^{5})\) 7a^5 + 7a^4 + 18a^3 + 11a^2 + 2a + 14 + (13a^5 + 5a^4 + a^3 + 16a^2 + 4a + 1)19 + (7a^5 + 7a^4 + 18a^3 + a^2 + 5a + 12)19^2 + (14a^5 + 12a^4 + 10a^2 + 3a + 14)19^3 + (14a^5 + 11a^4 + 7a^3 + 7a^2 + 10a + 18)19^4+O(19^5)
$r_{ 2 }$	$=$	\( 13 a^{5} + 9 a^{4} + 15 a^{3} + 7 a + \left(11 a^{5} + 9 a^{4} + 6 a^{3} + 6 a^{2} + 5\right)\cdot 19 + \left(13 a^{5} + 5 a^{4} + 9 a^{3} + a + 7\right)\cdot 19^{2} + \left(8 a^{5} + a^{4} + 12 a^{3} + 17 a^{2} + 12 a + 12\right)\cdot 19^{3} + \left(16 a^{5} + 17 a^{4} + 12 a^{3} + 18 a^{2} + a + 9\right)\cdot 19^{4} +O(19^{5})\) 13a^5 + 9a^4 + 15a^3 + 7a + (11a^5 + 9a^4 + 6a^3 + 6a^2 + 5)19 + (13a^5 + 5a^4 + 9a^3 + a + 7)19^2 + (8a^5 + a^4 + 12a^3 + 17a^2 + 12a + 12)19^3 + (16a^5 + 17a^4 + 12a^3 + 18a^2 + a + 9)*19^4+O(19^5)
$r_{ 3 }$	$=$	\( 8 a^{5} + 16 a^{4} + 4 a^{3} + 11 a^{2} + 4 a + 2 + \left(2 a^{5} + 13 a^{4} + 15 a^{3} + a^{2} + 10\right)\cdot 19 + \left(7 a^{5} + 9 a^{4} + 8 a^{3} + 15 a^{2} + 5 a + 8\right)\cdot 19^{2} + \left(4 a^{5} + 10 a^{4} + 7 a^{3} + 15 a^{2} + 2 a + 14\right)\cdot 19^{3} + \left(a^{5} + a^{4} + 6 a^{3} + 14 a^{2} + 6 a + 14\right)\cdot 19^{4} +O(19^{5})\) 8a^5 + 16a^4 + 4a^3 + 11a^2 + 4a + 2 + (2a^5 + 13a^4 + 15a^3 + a^2 + 10)19 + (7a^5 + 9a^4 + 8a^3 + 15a^2 + 5a + 8)19^2 + (4a^5 + 10a^4 + 7a^3 + 15a^2 + 2a + 14)19^3 + (a^5 + a^4 + 6a^3 + 14a^2 + 6a + 14)*19^4+O(19^5)
$r_{ 4 }$	$=$	\( 2 a^{5} + 8 a^{4} + 11 a^{3} + 5 a^{2} + 2 a + 1 + \left(15 a^{5} + 10 a^{4} + 5 a^{3} + 17 a^{2} + 15 a + 13\right)\cdot 19 + \left(9 a^{5} + 2 a^{4} + a^{3} + 18 a^{2} + 14 a + 9\right)\cdot 19^{2} + \left(5 a^{3} + 6 a^{2} + 4 a + 15\right)\cdot 19^{3} + \left(5 a^{5} + 3 a^{4} + 11 a^{3} + 9 a^{2} + 2 a + 18\right)\cdot 19^{4} +O(19^{5})\) 2a^5 + 8a^4 + 11a^3 + 5a^2 + 2a + 1 + (15a^5 + 10a^4 + 5a^3 + 17a^2 + 15a + 13)19 + (9a^5 + 2a^4 + a^3 + 18a^2 + 14a + 9)19^2 + (5a^3 + 6a^2 + 4a + 15)19^3 + (5a^5 + 3a^4 + 11a^3 + 9a^2 + 2a + 18)19^4+O(19^5)
$r_{ 5 }$	$=$	\( 11 a^{5} + 11 a^{4} + 12 a^{3} + a^{2} + 14 a + 3 + \left(8 a^{5} + 9 a^{4} + 17 a^{3} + 15 a^{2} + a + 13\right)\cdot 19 + \left(18 a^{5} + 9 a^{4} + 15 a^{3} + 15 a^{2} + 17 a + 18\right)\cdot 19^{2} + \left(5 a^{5} + 5 a^{4} + 6 a^{3} + 5 a^{2} + 15 a + 10\right)\cdot 19^{3} + \left(2 a^{4} + 18 a^{3} + 6 a + 12\right)\cdot 19^{4} +O(19^{5})\) 11a^5 + 11a^4 + 12a^3 + a^2 + 14a + 3 + (8a^5 + 9a^4 + 17a^3 + 15a^2 + a + 13)19 + (18a^5 + 9a^4 + 15a^3 + 15a^2 + 17a + 18)19^2 + (5a^5 + 5a^4 + 6a^3 + 5a^2 + 15a + 10)19^3 + (2a^4 + 18a^3 + 6a + 12)*19^4+O(19^5)
$r_{ 6 }$	$=$	\( 15 a^{5} + 6 a^{4} + 10 a^{3} + 11 a + \left(10 a^{5} + 11 a^{4} + 6 a^{3} + 4 a^{2} + 12 a + 16\right)\cdot 19 + \left(15 a^{4} + 6 a^{3} + 16 a^{2} + 4 a + 11\right)\cdot 19^{2} + \left(5 a^{5} + 5 a^{4} + 9 a^{3} + 13 a^{2} + 7 a + 2\right)\cdot 19^{3} + \left(12 a^{5} + 15 a^{4} + 14 a^{3} + 9 a^{2} + 16 a\right)\cdot 19^{4} +O(19^{5})\) 15a^5 + 6a^4 + 10a^3 + 11a + (10a^5 + 11a^4 + 6a^3 + 4a^2 + 12a + 16)19 + (15a^4 + 6a^3 + 16a^2 + 4a + 11)19^2 + (5a^5 + 5a^4 + 9a^3 + 13a^2 + 7a + 2)19^3 + (12a^5 + 15a^4 + 14a^3 + 9a^2 + 16a)*19^4+O(19^5)
$r_{ 7 }$	$=$	\( 10 a^{5} + 13 a^{4} + 13 a^{3} + 18 a^{2} + 4 a + 7 + \left(7 a^{5} + 4 a^{4} + 6 a^{3} + 8 a^{2} + 2 a + 11\right)\cdot 19 + \left(12 a^{5} + 5 a^{4} + 12 a^{3} + 4 a^{2} + 10 a + 1\right)\cdot 19^{2} + \left(5 a^{5} + 15 a^{4} + 17 a^{3} + 8 a^{2} + 12 a + 3\right)\cdot 19^{3} + \left(17 a^{5} + 3 a^{4} + 15 a^{3} + 2 a^{2} + 7 a + 2\right)\cdot 19^{4} +O(19^{5})\) 10a^5 + 13a^4 + 13a^3 + 18a^2 + 4a + 7 + (7a^5 + 4a^4 + 6a^3 + 8a^2 + 2a + 11)19 + (12a^5 + 5a^4 + 12a^3 + 4a^2 + 10a + 1)19^2 + (5a^5 + 15a^4 + 17a^3 + 8a^2 + 12a + 3)19^3 + (17a^5 + 3a^4 + 15a^3 + 2a^2 + 7a + 2)*19^4+O(19^5)
$r_{ 8 }$	$=$	\( 13 a^{5} + 15 a^{4} + 13 a^{3} + 16 a^{2} + 17 a + 13 + \left(15 a^{5} + 16 a^{4} + 8 a^{3} + 14 a^{2} + a + 3\right)\cdot 19 + \left(10 a^{5} + 15 a^{4} + 5 a^{2} + 2 a + 13\right)\cdot 19^{2} + \left(15 a^{5} + 12 a^{4} + 15 a^{3} + 17 a^{2} + 5 a + 2\right)\cdot 19^{3} + \left(3 a^{5} + 9 a^{4} + 10 a^{3} + 15 a^{2} + 5 a + 5\right)\cdot 19^{4} +O(19^{5})\) 13a^5 + 15a^4 + 13a^3 + 16a^2 + 17a + 13 + (15a^5 + 16a^4 + 8a^3 + 14a^2 + a + 3)19 + (10a^5 + 15a^4 + 5a^2 + 2a + 13)19^2 + (15a^5 + 12a^4 + 15a^3 + 17a^2 + 5a + 2)19^3 + (3a^5 + 9a^4 + 10a^3 + 15a^2 + 5a + 5)*19^4+O(19^5)
$r_{ 9 }$	$=$	\( 14 a^{5} + 18 a^{4} + a^{3} + 16 a^{2} + 14 a + 7 + \left(2 a^{5} + 2 a^{4} + 14 a^{3} + 5 a^{2} + 2 a + 11\right)\cdot 19 + \left(5 a^{5} + 2 a^{4} + 17 a^{3} + 8 a^{2} + 2 a + 4\right)\cdot 19^{2} + \left(9 a^{5} + 17 a^{4} + 17 a^{3} + 16 a^{2} + 3\right)\cdot 19^{3} + \left(18 a^{5} + 10 a^{4} + 3 a^{3} + 3 a^{2} + 3 a + 15\right)\cdot 19^{4} +O(19^{5})\) 14a^5 + 18a^4 + a^3 + 16a^2 + 14a + 7 + (2a^5 + 2a^4 + 14a^3 + 5a^2 + 2a + 11)19 + (5a^5 + 2a^4 + 17a^3 + 8a^2 + 2a + 4)19^2 + (9a^5 + 17a^4 + 17a^3 + 16a^2 + 3)19^3 + (18a^5 + 10a^4 + 3a^3 + 3a^2 + 3a + 15)*19^4+O(19^5)
$r_{ 10 }$	$=$	\( 12 a^{5} + 5 a^{4} + 6 a^{3} + 7 a^{2} + 6 a + 3 + \left(12 a^{5} + 10 a^{4} + 8 a^{3} + 12 a^{2} + 14 a + 3\right)\cdot 19 + \left(2 a^{5} + a^{4} + 15 a^{3} + 6 a^{2} + 4 a + 13\right)\cdot 19^{2} + \left(12 a^{5} + 15 a^{4} + 5 a^{3} + 16 a^{2} + 8\right)\cdot 19^{3} + \left(18 a^{5} + 11 a^{4} + 18 a^{3} + 2 a^{2} + 11 a + 2\right)\cdot 19^{4} +O(19^{5})\) 12a^5 + 5a^4 + 6a^3 + 7a^2 + 6a + 3 + (12a^5 + 10a^4 + 8a^3 + 12a^2 + 14a + 3)19 + (2a^5 + a^4 + 15a^3 + 6a^2 + 4a + 13)19^2 + (12a^5 + 15a^4 + 5a^3 + 16a^2 + 8)19^3 + (18a^5 + 11a^4 + 18a^3 + 2a^2 + 11a + 2)*19^4+O(19^5)
$r_{ 11 }$	$=$	\( 13 a^{5} + 15 a^{4} + 15 a^{3} + 12 a^{2} + 9 a + 11 + \left(12 a^{5} + 18 a^{4} + 13 a^{3} + 9 a^{2} + 18 a + 13\right)\cdot 19 + \left(3 a^{5} + 7 a^{4} + 11 a^{3} + 6 a^{2} + 17 a + 2\right)\cdot 19^{2} + \left(12 a^{5} + 9 a^{4} + 13 a^{3} + 16 a^{2} + 7 a + 11\right)\cdot 19^{3} + \left(17 a^{5} + a^{4} + 18 a^{2} + 12 a + 3\right)\cdot 19^{4} +O(19^{5})\) 13a^5 + 15a^4 + 15a^3 + 12a^2 + 9a + 11 + (12a^5 + 18a^4 + 13a^3 + 9a^2 + 18a + 13)19 + (3a^5 + 7a^4 + 11a^3 + 6a^2 + 17a + 2)19^2 + (12a^5 + 9a^4 + 13a^3 + 16a^2 + 7a + 11)19^3 + (17a^5 + a^4 + 18a^2 + 12a + 3)*19^4+O(19^5)
$r_{ 12 }$	$=$	\( 15 a^{5} + 10 a^{4} + 15 a^{3} + 17 a^{2} + 5 a + 15 + \left(8 a^{3} + a^{2} + 2 a + 11\right)\cdot 19 + \left(3 a^{5} + 12 a^{4} + 15 a^{3} + 14 a^{2} + 10 a + 10\right)\cdot 19^{2} + \left(a^{5} + 8 a^{4} + a^{3} + 7 a^{2} + 4 a + 14\right)\cdot 19^{3} + \left(7 a^{5} + 6 a^{4} + 13 a^{3} + 9 a^{2} + 12 a + 10\right)\cdot 19^{4} +O(19^{5})\) 15a^5 + 10a^4 + 15a^3 + 17a^2 + 5a + 15 + (8a^3 + a^2 + 2a + 11)19 + (3a^5 + 12a^4 + 15a^3 + 14a^2 + 10a + 10)19^2 + (a^5 + 8a^4 + a^3 + 7a^2 + 4a + 14)19^3 + (7a^5 + 6a^4 + 13a^3 + 9a^2 + 12a + 10)19^4+O(19^5)