LMFDB - Artin representation 1.99.15t1.a.a

Basic invariants

Dimension:	$1$
Group:	$C_{15}$
Conductor:	$99$$\medspace = 3^{2} \cdot 11 $
Artin field:	Galois closure of 15.15.10943023107606534329121.1
Galois orbit size:	$8$
Smallest permutation container:	$C_{15}$
Parity:	even
Dirichlet character:	$\chi_{99}(70,\cdot)$
Projective image:	$C_1$
Projective field:	Galois closure of $\Q$

Defining polynomial

$f(x)$

$=$

$ x^{15} - 27 x^{13} - 4 x^{12} + 252 x^{11} + 60 x^{10} - 976 x^{9} - 288 x^{8} + 1473 x^{7} + 384 x^{6} + \cdots - 1 $

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 6.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $ x^{5} + x + 14 $ x^5 + x + 14

Roots:

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

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

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

Character values on conjugacy classes

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

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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	1.99.15t1.a.a
Dimension	$1$
Group	$C_{15}$
Conductor	$99$
Root number	not computed
Indicator	$0$

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