Properties

 Label 1.135.18t1.b.b Dimension $1$ Group $C_{18}$ Conductor $135$ Root number not computed Indicator $0$

Related objects

Basic invariants

 Dimension: $1$ Group: $C_{18}$ Conductor: $$135$$$$\medspace = 3^{3} \cdot 5$$ Artin field: Galois closure of 18.0.5770142004982097067662109375.1 Galois orbit size: $6$ Smallest permutation container: $C_{18}$ Parity: odd Dirichlet character: $$\chi_{135}(29,\cdot)$$ Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

Defining polynomial

 $f(x)$ $=$ $$x^{18} + 18 x^{16} + 135 x^{14} + 546 x^{12} + 1287 x^{10} - 76 x^{9} + 1782 x^{8} - 684 x^{7} + 1386 x^{6} - 2052 x^{5} + 540 x^{4} - 2280 x^{3} + 81 x^{2} - 684 x + 5779$$ x^18 + 18*x^16 + 135*x^14 + 546*x^12 + 1287*x^10 - 76*x^9 + 1782*x^8 - 684*x^7 + 1386*x^6 - 2052*x^5 + 540*x^4 - 2280*x^3 + 81*x^2 - 684*x + 5779 .

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 8.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $$x^{9} + 3x^{3} + 8x^{2} + 9x + 18$$

Roots:
 $r_{ 1 }$ $=$ $$2 a^{8} + 19 a^{7} + 4 a^{6} + 11 a^{5} + 14 a^{3} + 22 a^{2} + 11 a + 17 + \left(6 a^{8} + 2 a^{7} + 14 a^{6} + 17 a^{5} + 15 a^{4} + 19 a^{3} + 9 a^{2} + a + 7\right)\cdot 23 + \left(20 a^{8} + 6 a^{7} + 4 a^{6} + 8 a^{5} + 19 a^{4} + 12 a^{3} + 17 a^{2} + 9 a + 20\right)\cdot 23^{2} + \left(16 a^{8} + 19 a^{7} + 21 a^{6} + 6 a^{5} + 8 a^{4} + 5 a^{3} + 18 a^{2} + 10 a\right)\cdot 23^{3} + \left(8 a^{8} + 3 a^{7} + 16 a^{6} + 4 a^{5} + 15 a^{4} + 7 a^{3} + 7 a^{2} + 2 a + 10\right)\cdot 23^{4} + \left(2 a^{8} + a^{7} + 4 a^{6} + 10 a^{5} + 22 a^{4} + 22 a^{3} + 7 a^{2} + 20\right)\cdot 23^{5} + \left(5 a^{8} + 16 a^{7} + 7 a^{6} + 6 a^{5} + 14 a^{4} + 16 a^{3} + 9 a^{2} + 6\right)\cdot 23^{6} + \left(16 a^{8} + 21 a^{7} + 8 a^{6} + 19 a^{5} + 7 a^{4} + 6 a^{2} + 11 a\right)\cdot 23^{7} +O(23^{8})$$ 2*a^8 + 19*a^7 + 4*a^6 + 11*a^5 + 14*a^3 + 22*a^2 + 11*a + 17 + (6*a^8 + 2*a^7 + 14*a^6 + 17*a^5 + 15*a^4 + 19*a^3 + 9*a^2 + a + 7)*23 + (20*a^8 + 6*a^7 + 4*a^6 + 8*a^5 + 19*a^4 + 12*a^3 + 17*a^2 + 9*a + 20)*23^2 + (16*a^8 + 19*a^7 + 21*a^6 + 6*a^5 + 8*a^4 + 5*a^3 + 18*a^2 + 10*a)*23^3 + (8*a^8 + 3*a^7 + 16*a^6 + 4*a^5 + 15*a^4 + 7*a^3 + 7*a^2 + 2*a + 10)*23^4 + (2*a^8 + a^7 + 4*a^6 + 10*a^5 + 22*a^4 + 22*a^3 + 7*a^2 + 20)*23^5 + (5*a^8 + 16*a^7 + 7*a^6 + 6*a^5 + 14*a^4 + 16*a^3 + 9*a^2 + 6)*23^6 + (16*a^8 + 21*a^7 + 8*a^6 + 19*a^5 + 7*a^4 + 6*a^2 + 11*a)*23^7+O(23^8) $r_{ 2 }$ $=$ $$10 a^{8} + 13 a^{7} + 4 a^{6} + 14 a^{5} + 15 a^{4} + 17 a^{3} + 19 a^{2} + 22 a + 13 + \left(18 a^{8} + 14 a^{7} + 7 a^{5} + 15 a^{4} + 14 a^{3} + 5 a^{2} + 4 a + 13\right)\cdot 23 + \left(18 a^{8} + 5 a^{6} + 5 a^{5} + 17 a^{4} + 14 a^{3} + 19 a^{2} + 7 a + 13\right)\cdot 23^{2} + \left(4 a^{8} + 17 a^{7} + 7 a^{6} + 19 a^{5} + 9 a^{4} + a^{3} + 12 a^{2} + 19 a + 5\right)\cdot 23^{3} + \left(7 a^{8} + 8 a^{7} + 13 a^{6} + 19 a^{5} + 17 a^{4} + 17 a^{3} + 17 a^{2} + 16 a + 3\right)\cdot 23^{4} + \left(19 a^{8} + 5 a^{7} + a^{6} + 18 a^{5} + 8 a^{4} + 18 a^{3} + 4 a^{2} + 2 a + 7\right)\cdot 23^{5} + \left(3 a^{8} + 12 a^{6} + 8 a^{5} + 18 a^{4} + 20 a^{3} + 13 a^{2} + 5 a + 10\right)\cdot 23^{6} + \left(17 a^{8} + 8 a^{7} + 20 a^{6} + a^{5} + 13 a^{4} + 6 a^{3} + 20 a^{2} + 12 a + 8\right)\cdot 23^{7} +O(23^{8})$$ 10*a^8 + 13*a^7 + 4*a^6 + 14*a^5 + 15*a^4 + 17*a^3 + 19*a^2 + 22*a + 13 + (18*a^8 + 14*a^7 + 7*a^5 + 15*a^4 + 14*a^3 + 5*a^2 + 4*a + 13)*23 + (18*a^8 + 5*a^6 + 5*a^5 + 17*a^4 + 14*a^3 + 19*a^2 + 7*a + 13)*23^2 + (4*a^8 + 17*a^7 + 7*a^6 + 19*a^5 + 9*a^4 + a^3 + 12*a^2 + 19*a + 5)*23^3 + (7*a^8 + 8*a^7 + 13*a^6 + 19*a^5 + 17*a^4 + 17*a^3 + 17*a^2 + 16*a + 3)*23^4 + (19*a^8 + 5*a^7 + a^6 + 18*a^5 + 8*a^4 + 18*a^3 + 4*a^2 + 2*a + 7)*23^5 + (3*a^8 + 12*a^6 + 8*a^5 + 18*a^4 + 20*a^3 + 13*a^2 + 5*a + 10)*23^6 + (17*a^8 + 8*a^7 + 20*a^6 + a^5 + 13*a^4 + 6*a^3 + 20*a^2 + 12*a + 8)*23^7+O(23^8) $r_{ 3 }$ $=$ $$18 a^{8} + 8 a^{7} + 10 a^{6} + 2 a^{5} + 2 a^{4} + 11 a^{3} + 10 a^{2} + 2 a + 17 + \left(4 a^{8} + 9 a^{7} + 16 a^{6} + 7 a^{5} + a^{4} + 20 a^{3} + 10 a^{2} + 18 a + 11\right)\cdot 23 + \left(19 a^{8} + 9 a^{7} + 16 a^{6} + a^{4} + 14 a^{3} + 2 a^{2} + 12 a + 10\right)\cdot 23^{2} + \left(15 a^{8} + 15 a^{7} + 4 a^{6} + a^{5} + 11 a^{4} + 13 a^{3} + 14 a^{2} + 9 a + 22\right)\cdot 23^{3} + \left(3 a^{8} + 22 a^{7} + 17 a^{6} + 7 a^{5} + 16 a^{4} + 8 a^{2} + a + 20\right)\cdot 23^{4} + \left(6 a^{8} + 4 a^{7} + 4 a^{6} + 4 a^{5} + 19 a^{4} + 3 a^{3} + 4 a^{2} + 10 a + 2\right)\cdot 23^{5} + \left(a^{8} + 14 a^{7} + 6 a^{6} + 15 a^{4} + 9 a^{3} + 5 a^{2} + 1\right)\cdot 23^{6} + \left(14 a^{8} + 21 a^{7} + 8 a^{6} + 18 a^{5} + 19 a^{4} + 2 a^{3} + 16 a^{2} + 10 a + 13\right)\cdot 23^{7} +O(23^{8})$$ 18*a^8 + 8*a^7 + 10*a^6 + 2*a^5 + 2*a^4 + 11*a^3 + 10*a^2 + 2*a + 17 + (4*a^8 + 9*a^7 + 16*a^6 + 7*a^5 + a^4 + 20*a^3 + 10*a^2 + 18*a + 11)*23 + (19*a^8 + 9*a^7 + 16*a^6 + a^4 + 14*a^3 + 2*a^2 + 12*a + 10)*23^2 + (15*a^8 + 15*a^7 + 4*a^6 + a^5 + 11*a^4 + 13*a^3 + 14*a^2 + 9*a + 22)*23^3 + (3*a^8 + 22*a^7 + 17*a^6 + 7*a^5 + 16*a^4 + 8*a^2 + a + 20)*23^4 + (6*a^8 + 4*a^7 + 4*a^6 + 4*a^5 + 19*a^4 + 3*a^3 + 4*a^2 + 10*a + 2)*23^5 + (a^8 + 14*a^7 + 6*a^6 + 15*a^4 + 9*a^3 + 5*a^2 + 1)*23^6 + (14*a^8 + 21*a^7 + 8*a^6 + 18*a^5 + 19*a^4 + 2*a^3 + 16*a^2 + 10*a + 13)*23^7+O(23^8) $r_{ 4 }$ $=$ $$19 a^{8} + 21 a^{7} + 14 a^{6} + 17 a^{5} + 20 a^{4} + 19 a^{2} + 10 a + 4 + \left(20 a^{8} + a^{7} + 19 a^{6} + 7 a^{5} + 22 a^{4} + 5 a^{3} + 9 a^{2} + 3 a + 8\right)\cdot 23 + \left(11 a^{8} + 21 a^{7} + 22 a^{6} + 17 a^{5} + 19 a^{4} + 10 a^{3} + 16 a^{2} + 21 a + 6\right)\cdot 23^{2} + \left(20 a^{8} + 20 a^{7} + 17 a^{6} + 22 a^{5} + 10 a^{4} + 19 a^{3} + 20 a^{2} + 22 a + 8\right)\cdot 23^{3} + \left(4 a^{8} + 16 a^{7} + 3 a^{6} + 9 a^{5} + 9 a^{3} + 19 a^{2} + 7 a + 11\right)\cdot 23^{4} + \left(22 a^{8} + 2 a^{7} + 7 a^{6} + 20 a^{5} + 7 a^{4} + 8 a^{3} + 9 a^{2} + 22 a + 22\right)\cdot 23^{5} + \left(18 a^{8} + 2 a^{7} + 22 a^{6} + 19 a^{5} + 15 a^{4} + 20 a^{3} + 17 a^{2} + 5 a + 22\right)\cdot 23^{6} + \left(a^{7} + 13 a^{6} + 17 a^{5} + 12 a^{4} + 10 a^{3} + 22 a^{2} + 8 a + 2\right)\cdot 23^{7} +O(23^{8})$$ 19*a^8 + 21*a^7 + 14*a^6 + 17*a^5 + 20*a^4 + 19*a^2 + 10*a + 4 + (20*a^8 + a^7 + 19*a^6 + 7*a^5 + 22*a^4 + 5*a^3 + 9*a^2 + 3*a + 8)*23 + (11*a^8 + 21*a^7 + 22*a^6 + 17*a^5 + 19*a^4 + 10*a^3 + 16*a^2 + 21*a + 6)*23^2 + (20*a^8 + 20*a^7 + 17*a^6 + 22*a^5 + 10*a^4 + 19*a^3 + 20*a^2 + 22*a + 8)*23^3 + (4*a^8 + 16*a^7 + 3*a^6 + 9*a^5 + 9*a^3 + 19*a^2 + 7*a + 11)*23^4 + (22*a^8 + 2*a^7 + 7*a^6 + 20*a^5 + 7*a^4 + 8*a^3 + 9*a^2 + 22*a + 22)*23^5 + (18*a^8 + 2*a^7 + 22*a^6 + 19*a^5 + 15*a^4 + 20*a^3 + 17*a^2 + 5*a + 22)*23^6 + (a^7 + 13*a^6 + 17*a^5 + 12*a^4 + 10*a^3 + 22*a^2 + 8*a + 2)*23^7+O(23^8) $r_{ 5 }$ $=$ $$17 a^{8} + 9 a^{7} + a^{6} + 10 a^{4} + 6 a^{3} + 4 a^{2} + 5 a + 10 + \left(19 a^{8} + 12 a^{7} + 2 a^{6} + 16 a^{5} + 7 a^{4} + 6 a^{3} + 16 a^{2} + 8 a + 1\right)\cdot 23 + \left(20 a^{8} + 6 a^{7} + 20 a^{6} + 20 a^{5} + 12 a^{4} + 5 a^{3} + 10 a^{2} + 18 a + 10\right)\cdot 23^{2} + \left(14 a^{8} + 12 a^{7} + 18 a^{6} + 17 a^{5} + 4 a^{4} + 20 a^{3} + a^{2} + 16 a + 3\right)\cdot 23^{3} + \left(6 a^{8} + 2 a^{7} + 15 a^{6} + 14 a^{5} + 21 a^{4} + 22 a^{3} + 17 a^{2} + 15 a + 11\right)\cdot 23^{4} + \left(12 a^{8} + 3 a^{7} + 5 a^{6} + 14 a^{5} + 19 a^{4} + 20 a^{3} + a^{2} + 3 a + 11\right)\cdot 23^{5} + \left(12 a^{8} + 8 a^{7} + 4 a^{6} + 8 a^{5} + 21 a^{4} + 22 a^{3} + 11 a^{2} + 4 a + 3\right)\cdot 23^{6} + \left(19 a^{8} + 22 a^{7} + 21 a^{6} + 14 a^{5} + 22 a^{4} + 4 a^{3} + 8 a^{2} + 14 a + 8\right)\cdot 23^{7} +O(23^{8})$$ 17*a^8 + 9*a^7 + a^6 + 10*a^4 + 6*a^3 + 4*a^2 + 5*a + 10 + (19*a^8 + 12*a^7 + 2*a^6 + 16*a^5 + 7*a^4 + 6*a^3 + 16*a^2 + 8*a + 1)*23 + (20*a^8 + 6*a^7 + 20*a^6 + 20*a^5 + 12*a^4 + 5*a^3 + 10*a^2 + 18*a + 10)*23^2 + (14*a^8 + 12*a^7 + 18*a^6 + 17*a^5 + 4*a^4 + 20*a^3 + a^2 + 16*a + 3)*23^3 + (6*a^8 + 2*a^7 + 15*a^6 + 14*a^5 + 21*a^4 + 22*a^3 + 17*a^2 + 15*a + 11)*23^4 + (12*a^8 + 3*a^7 + 5*a^6 + 14*a^5 + 19*a^4 + 20*a^3 + a^2 + 3*a + 11)*23^5 + (12*a^8 + 8*a^7 + 4*a^6 + 8*a^5 + 21*a^4 + 22*a^3 + 11*a^2 + 4*a + 3)*23^6 + (19*a^8 + 22*a^7 + 21*a^6 + 14*a^5 + 22*a^4 + 4*a^3 + 8*a^2 + 14*a + 8)*23^7+O(23^8) $r_{ 6 }$ $=$ $$9 a^{8} + 3 a^{7} + 2 a^{6} + 20 a^{5} + 6 a^{4} + 5 a^{3} + 7 a^{2} + 13 a + 18 + \left(19 a^{8} + 4 a^{7} + 14 a^{6} + 17 a^{5} + 21 a^{4} + 22 a^{3} + 6 a^{2} + 19 a + 14\right)\cdot 23 + \left(15 a^{8} + 5 a^{7} + 11 a^{6} + 22 a^{5} + 7 a^{4} + 10 a^{3} + 6 a^{2} + 5 a + 18\right)\cdot 23^{2} + \left(9 a^{8} + 22 a^{7} + 16 a^{6} + 2 a^{5} + 6 a^{4} + a^{3} + 13 a^{2} + 8 a + 18\right)\cdot 23^{3} + \left(13 a^{8} + 21 a^{7} + 22 a^{6} + 15 a^{5} + 11 a^{4} + 10 a^{3} + 7 a^{2} + 9 a + 5\right)\cdot 23^{4} + \left(3 a^{8} + 7 a^{7} + a^{6} + 13 a^{5} + 14 a^{4} + 21 a^{3} + 4 a^{2} + 16 a + 18\right)\cdot 23^{5} + \left(6 a^{8} + 18 a^{7} + 19 a^{6} + 19 a^{5} + 20 a^{4} + 4 a^{3} + 6 a^{2} + 14 a + 4\right)\cdot 23^{6} + \left(18 a^{8} + 5 a^{6} + 14 a^{5} + 4 a^{4} + 18 a^{3} + 18 a^{2} + 20 a + 22\right)\cdot 23^{7} +O(23^{8})$$ 9*a^8 + 3*a^7 + 2*a^6 + 20*a^5 + 6*a^4 + 5*a^3 + 7*a^2 + 13*a + 18 + (19*a^8 + 4*a^7 + 14*a^6 + 17*a^5 + 21*a^4 + 22*a^3 + 6*a^2 + 19*a + 14)*23 + (15*a^8 + 5*a^7 + 11*a^6 + 22*a^5 + 7*a^4 + 10*a^3 + 6*a^2 + 5*a + 18)*23^2 + (9*a^8 + 22*a^7 + 16*a^6 + 2*a^5 + 6*a^4 + a^3 + 13*a^2 + 8*a + 18)*23^3 + (13*a^8 + 21*a^7 + 22*a^6 + 15*a^5 + 11*a^4 + 10*a^3 + 7*a^2 + 9*a + 5)*23^4 + (3*a^8 + 7*a^7 + a^6 + 13*a^5 + 14*a^4 + 21*a^3 + 4*a^2 + 16*a + 18)*23^5 + (6*a^8 + 18*a^7 + 19*a^6 + 19*a^5 + 20*a^4 + 4*a^3 + 6*a^2 + 14*a + 4)*23^6 + (18*a^8 + 5*a^6 + 14*a^5 + 4*a^4 + 18*a^3 + 18*a^2 + 20*a + 22)*23^7+O(23^8) $r_{ 7 }$ $=$ $$5 a^{8} + 15 a^{7} + 9 a^{6} + 8 a^{5} + 3 a^{4} + 15 a^{3} + 13 a^{2} + 13 a + 21 + \left(10 a^{8} + 9 a^{7} + 8 a^{6} + 14 a^{5} + 12 a^{4} + 11 a^{3} + a^{2} + 13 a + 12\right)\cdot 23 + \left(7 a^{8} + 11 a^{7} + 18 a^{6} + 19 a^{5} + 22 a^{4} + 12 a^{2} + 9 a + 16\right)\cdot 23^{2} + \left(21 a^{8} + 5 a^{7} + 4 a^{6} + 17 a^{5} + 22 a^{4} + 13 a^{3} + 7 a^{2} + a + 22\right)\cdot 23^{3} + \left(11 a^{8} + 6 a^{7} + 13 a^{6} + 6 a^{5} + 19 a^{4} + 19 a^{3} + 3 a^{2} + 5 a + 14\right)\cdot 23^{4} + \left(5 a^{8} + 4 a^{7} + 6 a^{6} + 18 a^{5} + 22 a^{4} + 10 a^{3} + 14 a^{2} + 2 a + 4\right)\cdot 23^{5} + \left(19 a^{8} + 22 a^{7} + 18 a^{6} + 14 a^{5} + 14 a^{4} + 19 a^{3} + 6 a^{2} + 5 a + 14\right)\cdot 23^{6} + \left(3 a^{8} + 8 a^{7} + 6 a^{6} + 21 a^{4} + 8 a^{3} + 6 a^{2} + 18 a + 10\right)\cdot 23^{7} +O(23^{8})$$ 5*a^8 + 15*a^7 + 9*a^6 + 8*a^5 + 3*a^4 + 15*a^3 + 13*a^2 + 13*a + 21 + (10*a^8 + 9*a^7 + 8*a^6 + 14*a^5 + 12*a^4 + 11*a^3 + a^2 + 13*a + 12)*23 + (7*a^8 + 11*a^7 + 18*a^6 + 19*a^5 + 22*a^4 + 12*a^2 + 9*a + 16)*23^2 + (21*a^8 + 5*a^7 + 4*a^6 + 17*a^5 + 22*a^4 + 13*a^3 + 7*a^2 + a + 22)*23^3 + (11*a^8 + 6*a^7 + 13*a^6 + 6*a^5 + 19*a^4 + 19*a^3 + 3*a^2 + 5*a + 14)*23^4 + (5*a^8 + 4*a^7 + 6*a^6 + 18*a^5 + 22*a^4 + 10*a^3 + 14*a^2 + 2*a + 4)*23^5 + (19*a^8 + 22*a^7 + 18*a^6 + 14*a^5 + 14*a^4 + 19*a^3 + 6*a^2 + 5*a + 14)*23^6 + (3*a^8 + 8*a^7 + 6*a^6 + 21*a^4 + 8*a^3 + 6*a^2 + 18*a + 10)*23^7+O(23^8) $r_{ 8 }$ $=$ $$22 a^{8} + 10 a^{7} + 21 a^{5} + 8 a^{3} + 14 a^{2} + 21 + \left(14 a^{8} + 11 a^{7} + 18 a^{6} + 11 a^{4} + 6 a^{3} + 11 a^{2} + 6 a + 1\right)\cdot 23 + \left(3 a^{8} + 13 a^{7} + 4 a^{6} + 9 a^{5} + 3 a^{4} + 12 a^{3} + 17 a^{2} + 15 a\right)\cdot 23^{2} + \left(4 a^{8} + 19 a^{7} + 5 a^{5} + 12 a^{4} + 13 a^{3} + 17 a^{2} + 21 a + 15\right)\cdot 23^{3} + \left(6 a^{8} + 22 a^{7} + 6 a^{6} + 6 a^{5} + 2 a^{4} + 16 a^{3} + 22 a^{2} + 9 a + 19\right)\cdot 23^{4} + \left(18 a^{8} + 15 a^{7} + 9 a^{6} + 7 a^{5} + 16 a^{4} + 3 a^{3} + 21 a^{2} + 21 a + 18\right)\cdot 23^{5} + \left(7 a^{8} + 21 a^{7} + 5 a^{6} + 11 a^{5} + 15 a^{4} + 6 a^{3} + 21 a^{2} + 11 a + 8\right)\cdot 23^{6} + \left(18 a^{8} + 12 a^{7} + 2 a^{6} + 4 a^{5} + 11 a^{4} + 3 a^{3} + 16 a^{2} + 19 a + 9\right)\cdot 23^{7} +O(23^{8})$$ 22*a^8 + 10*a^7 + 21*a^5 + 8*a^3 + 14*a^2 + 21 + (14*a^8 + 11*a^7 + 18*a^6 + 11*a^4 + 6*a^3 + 11*a^2 + 6*a + 1)*23 + (3*a^8 + 13*a^7 + 4*a^6 + 9*a^5 + 3*a^4 + 12*a^3 + 17*a^2 + 15*a)*23^2 + (4*a^8 + 19*a^7 + 5*a^5 + 12*a^4 + 13*a^3 + 17*a^2 + 21*a + 15)*23^3 + (6*a^8 + 22*a^7 + 6*a^6 + 6*a^5 + 2*a^4 + 16*a^3 + 22*a^2 + 9*a + 19)*23^4 + (18*a^8 + 15*a^7 + 9*a^6 + 7*a^5 + 16*a^4 + 3*a^3 + 21*a^2 + 21*a + 18)*23^5 + (7*a^8 + 21*a^7 + 5*a^6 + 11*a^5 + 15*a^4 + 6*a^3 + 21*a^2 + 11*a + 8)*23^6 + (18*a^8 + 12*a^7 + 2*a^6 + 4*a^5 + 11*a^4 + 3*a^3 + 16*a^2 + 19*a + 9)*23^7+O(23^8) $r_{ 9 }$ $=$ $$2 a^{8} + 16 a^{7} + 18 a^{6} + 21 a^{5} + 7 a^{4} + a^{2} + 14 a + 11 + \left(18 a^{8} + 11 a^{7} + 3 a^{5} + 16 a^{4} + 3 a^{3} + 4 a^{2} + 22 a + 9\right)\cdot 23 + \left(21 a^{7} + 18 a^{6} + 5 a^{5} + 14 a^{3} + 5 a^{2} + 12 a + 2\right)\cdot 23^{2} + \left(10 a^{8} + 6 a^{7} + 16 a^{6} + 6 a^{5} + 20 a^{4} + 14 a^{3} + 13 a^{2} + 8 a + 1\right)\cdot 23^{3} + \left(8 a^{7} + 15 a^{6} + 6 a^{5} + a^{3} + 20 a^{2} + 4 a + 15\right)\cdot 23^{4} + \left(10 a^{8} + 3 a^{7} + 17 a^{6} + 6 a^{5} + 9 a^{4} + 22 a^{3} + 12 a^{2} + 16 a + 8\right)\cdot 23^{5} + \left(11 a^{8} + 4 a^{7} + 18 a^{6} + 14 a^{5} + 9 a^{4} + a^{3} + 9 a + 19\right)\cdot 23^{6} + \left(5 a^{8} + 12 a^{7} + 12 a^{6} + 22 a^{5} + 16 a^{4} + 9 a^{3} + 10 a^{2} + 19 a + 4\right)\cdot 23^{7} +O(23^{8})$$ 2*a^8 + 16*a^7 + 18*a^6 + 21*a^5 + 7*a^4 + a^2 + 14*a + 11 + (18*a^8 + 11*a^7 + 3*a^5 + 16*a^4 + 3*a^3 + 4*a^2 + 22*a + 9)*23 + (21*a^7 + 18*a^6 + 5*a^5 + 14*a^3 + 5*a^2 + 12*a + 2)*23^2 + (10*a^8 + 6*a^7 + 16*a^6 + 6*a^5 + 20*a^4 + 14*a^3 + 13*a^2 + 8*a + 1)*23^3 + (8*a^7 + 15*a^6 + 6*a^5 + a^3 + 20*a^2 + 4*a + 15)*23^4 + (10*a^8 + 3*a^7 + 17*a^6 + 6*a^5 + 9*a^4 + 22*a^3 + 12*a^2 + 16*a + 8)*23^5 + (11*a^8 + 4*a^7 + 18*a^6 + 14*a^5 + 9*a^4 + a^3 + 9*a + 19)*23^6 + (5*a^8 + 12*a^7 + 12*a^6 + 22*a^5 + 16*a^4 + 9*a^3 + 10*a^2 + 19*a + 4)*23^7+O(23^8) $r_{ 10 }$ $=$ $$5 a^{8} + 21 a^{7} + 7 a^{6} + 11 a^{5} + 10 a^{4} + 13 a^{3} + 15 a^{2} + 5 a + 16 + \left(12 a^{8} + 20 a^{7} + 5 a^{6} + 22 a^{5} + 9 a^{4} + 9 a^{3} + a^{2} + 20 a + 18\right)\cdot 23 + \left(5 a^{8} + 6 a^{7} + 12 a^{6} + 17 a^{5} + 12 a^{4} + 11 a^{3} + 18 a^{2} + 21 a + 1\right)\cdot 23^{2} + \left(6 a^{8} + 17 a^{7} + 2 a^{6} + 8 a^{5} + a^{3} + 8 a^{2} + 20 a + 22\right)\cdot 23^{3} + \left(6 a^{8} + 22 a^{7} + 11 a^{6} + 5 a^{5} + 15 a^{4} + 9 a^{2} + 8 a + 6\right)\cdot 23^{4} + \left(2 a^{7} + 4 a^{6} + 3 a^{5} + 6 a^{4} + 13 a^{3} + 19 a^{2} + 4 a + 9\right)\cdot 23^{5} + \left(3 a^{8} + 14 a^{7} + 4 a^{6} + 3 a^{5} + 10 a^{4} + 20 a^{3} + a^{2} + 20 a + 5\right)\cdot 23^{6} + \left(18 a^{8} + 14 a^{7} + 2 a^{6} + 2 a^{5} + 9 a^{4} + 18 a^{3} + 10 a^{2} + 21 a + 5\right)\cdot 23^{7} +O(23^{8})$$ 5*a^8 + 21*a^7 + 7*a^6 + 11*a^5 + 10*a^4 + 13*a^3 + 15*a^2 + 5*a + 16 + (12*a^8 + 20*a^7 + 5*a^6 + 22*a^5 + 9*a^4 + 9*a^3 + a^2 + 20*a + 18)*23 + (5*a^8 + 6*a^7 + 12*a^6 + 17*a^5 + 12*a^4 + 11*a^3 + 18*a^2 + 21*a + 1)*23^2 + (6*a^8 + 17*a^7 + 2*a^6 + 8*a^5 + a^3 + 8*a^2 + 20*a + 22)*23^3 + (6*a^8 + 22*a^7 + 11*a^6 + 5*a^5 + 15*a^4 + 9*a^2 + 8*a + 6)*23^4 + (2*a^7 + 4*a^6 + 3*a^5 + 6*a^4 + 13*a^3 + 19*a^2 + 4*a + 9)*23^5 + (3*a^8 + 14*a^7 + 4*a^6 + 3*a^5 + 10*a^4 + 20*a^3 + a^2 + 20*a + 5)*23^6 + (18*a^8 + 14*a^7 + 2*a^6 + 2*a^5 + 9*a^4 + 18*a^3 + 10*a^2 + 21*a + 5)*23^7+O(23^8) $r_{ 11 }$ $=$ $$20 a^{8} + 11 a^{7} + 20 a^{6} + 3 a^{5} + 7 a^{4} + 12 a^{3} + 12 a^{2} + 5 a + 18 + \left(6 a^{8} + 6 a^{7} + 6 a^{6} + 12 a^{5} + 17 a^{4} + 17 a^{3} + 18 a + 6\right)\cdot 23 + \left(9 a^{8} + 11 a^{7} + 14 a^{6} + 2 a^{5} + 2 a^{4} + 6 a^{3} + 6 a^{2} + 21 a + 17\right)\cdot 23^{2} + \left(21 a^{8} + 11 a^{7} + 10 a^{6} + 2 a^{5} + 12 a^{4} + a^{3} + 8 a^{2} + 20 a\right)\cdot 23^{3} + \left(2 a^{8} + 21 a^{7} + 7 a^{6} + 16 a^{5} + 13 a^{4} + 13 a^{3} + 21 a^{2} + 20 a + 6\right)\cdot 23^{4} + \left(7 a^{8} + 11 a^{7} + 15 a^{6} + 17 a^{5} + 11 a^{4} + 3 a^{3} + 16 a^{2} + 2 a + 16\right)\cdot 23^{5} + \left(4 a^{8} + 19 a^{7} + 22 a^{6} + 17 a^{5} + 3 a^{4} + 18 a^{3} + 5 a^{2} + 4 a + 14\right)\cdot 23^{6} + \left(8 a^{8} + 22 a^{7} + 18 a^{6} + 16 a^{5} + 18 a^{4} + 22 a^{3} + 19 a^{2} + 11 a + 15\right)\cdot 23^{7} +O(23^{8})$$ 20*a^8 + 11*a^7 + 20*a^6 + 3*a^5 + 7*a^4 + 12*a^3 + 12*a^2 + 5*a + 18 + (6*a^8 + 6*a^7 + 6*a^6 + 12*a^5 + 17*a^4 + 17*a^3 + 18*a + 6)*23 + (9*a^8 + 11*a^7 + 14*a^6 + 2*a^5 + 2*a^4 + 6*a^3 + 6*a^2 + 21*a + 17)*23^2 + (21*a^8 + 11*a^7 + 10*a^6 + 2*a^5 + 12*a^4 + a^3 + 8*a^2 + 20*a)*23^3 + (2*a^8 + 21*a^7 + 7*a^6 + 16*a^5 + 13*a^4 + 13*a^3 + 21*a^2 + 20*a + 6)*23^4 + (7*a^8 + 11*a^7 + 15*a^6 + 17*a^5 + 11*a^4 + 3*a^3 + 16*a^2 + 2*a + 16)*23^5 + (4*a^8 + 19*a^7 + 22*a^6 + 17*a^5 + 3*a^4 + 18*a^3 + 5*a^2 + 4*a + 14)*23^6 + (8*a^8 + 22*a^7 + 18*a^6 + 16*a^5 + 18*a^4 + 22*a^3 + 19*a^2 + 11*a + 15)*23^7+O(23^8) $r_{ 12 }$ $=$ $$19 a^{8} + 10 a^{7} + 10 a^{6} + 11 a^{4} + 19 a^{3} + 17 a + 17 + \left(a^{8} + 20 a^{7} + 22 a^{6} + 22 a^{5} + 20 a^{4} + 9 a^{3} + a^{2} + 6 a + 7\right)\cdot 23 + \left(22 a^{8} + 4 a^{7} + 9 a^{6} + 4 a^{5} + 10 a^{4} + 15 a^{3} + 8 a^{2} + 15 a + 12\right)\cdot 23^{2} + \left(a^{8} + 7 a^{7} + 11 a^{6} + 4 a^{5} + 14 a^{4} + 22 a^{3} + 21 a^{2} + 8 a + 19\right)\cdot 23^{3} + \left(3 a^{8} + 11 a^{7} + 9 a^{6} + a^{5} + 8 a^{4} + 2 a^{3} + 11 a^{2} + 12 a + 11\right)\cdot 23^{4} + \left(12 a^{8} + 6 a^{7} + 22 a^{6} + 4 a^{3} + 22 a^{2} + 3 a + 8\right)\cdot 23^{5} + \left(8 a^{8} + 5 a^{7} + 12 a^{6} + 8 a^{5} + 15 a^{4} + 22 a^{3} + 4 a^{2} + 7 a + 4\right)\cdot 23^{6} + \left(8 a^{8} + 17 a^{7} + 18 a^{6} + 4 a^{5} + 16 a^{4} + 6 a^{3} + 6 a^{2} + 22\right)\cdot 23^{7} +O(23^{8})$$ 19*a^8 + 10*a^7 + 10*a^6 + 11*a^4 + 19*a^3 + 17*a + 17 + (a^8 + 20*a^7 + 22*a^6 + 22*a^5 + 20*a^4 + 9*a^3 + a^2 + 6*a + 7)*23 + (22*a^8 + 4*a^7 + 9*a^6 + 4*a^5 + 10*a^4 + 15*a^3 + 8*a^2 + 15*a + 12)*23^2 + (a^8 + 7*a^7 + 11*a^6 + 4*a^5 + 14*a^4 + 22*a^3 + 21*a^2 + 8*a + 19)*23^3 + (3*a^8 + 11*a^7 + 9*a^6 + a^5 + 8*a^4 + 2*a^3 + 11*a^2 + 12*a + 11)*23^4 + (12*a^8 + 6*a^7 + 22*a^6 + 4*a^3 + 22*a^2 + 3*a + 8)*23^5 + (8*a^8 + 5*a^7 + 12*a^6 + 8*a^5 + 15*a^4 + 22*a^3 + 4*a^2 + 7*a + 4)*23^6 + (8*a^8 + 17*a^7 + 18*a^6 + 4*a^5 + 16*a^4 + 6*a^3 + 6*a^2 + 22)*23^7+O(23^8) $r_{ 13 }$ $=$ $$16 a^{8} + 19 a^{7} + 19 a^{6} + 11 a^{5} + 19 a^{4} + 17 a^{3} + 3 a^{2} + 15 a + 21 + \left(10 a^{8} + a^{7} + 12 a^{6} + 9 a^{5} + 13 a^{4} + 3 a^{3} + 13 a^{2} + 7 a + 22\right)\cdot 23 + \left(3 a^{7} + 20 a^{6} + 2 a^{5} + 7 a^{4} + a^{3} + 8 a^{2} + 22 a + 17\right)\cdot 23^{2} + \left(8 a^{8} + 15 a^{7} + 22 a^{6} + 10 a^{5} + a^{4} + 2 a^{3} + 2 a^{2} + 3 a + 4\right)\cdot 23^{3} + \left(8 a^{8} + 18 a^{7} + 8 a^{6} + 2 a^{5} + 14 a^{4} + 13 a^{3} + 7 a^{2} + 10 a + 9\right)\cdot 23^{4} + \left(9 a^{8} + 17 a^{7} + a^{6} + 7 a^{5} + 22 a^{4} + 9 a^{3} + 4 a^{2} + 20 a + 22\right)\cdot 23^{5} + \left(9 a^{8} + 10 a^{7} + 20 a^{6} + 4 a^{5} + 12 a^{4} + 3 a^{3} + 2 a^{2} + 20 a\right)\cdot 23^{6} + \left(14 a^{8} + 16 a^{7} + a^{6} + 21 a^{5} + 20 a^{4} + 15 a^{3} + 9 a^{2} + 2 a + 17\right)\cdot 23^{7} +O(23^{8})$$ 16*a^8 + 19*a^7 + 19*a^6 + 11*a^5 + 19*a^4 + 17*a^3 + 3*a^2 + 15*a + 21 + (10*a^8 + a^7 + 12*a^6 + 9*a^5 + 13*a^4 + 3*a^3 + 13*a^2 + 7*a + 22)*23 + (3*a^7 + 20*a^6 + 2*a^5 + 7*a^4 + a^3 + 8*a^2 + 22*a + 17)*23^2 + (8*a^8 + 15*a^7 + 22*a^6 + 10*a^5 + a^4 + 2*a^3 + 2*a^2 + 3*a + 4)*23^3 + (8*a^8 + 18*a^7 + 8*a^6 + 2*a^5 + 14*a^4 + 13*a^3 + 7*a^2 + 10*a + 9)*23^4 + (9*a^8 + 17*a^7 + a^6 + 7*a^5 + 22*a^4 + 9*a^3 + 4*a^2 + 20*a + 22)*23^5 + (9*a^8 + 10*a^7 + 20*a^6 + 4*a^5 + 12*a^4 + 3*a^3 + 2*a^2 + 20*a)*23^6 + (14*a^8 + 16*a^7 + a^6 + 21*a^5 + 20*a^4 + 15*a^3 + 9*a^2 + 2*a + 17)*23^7+O(23^8) $r_{ 14 }$ $=$ $$19 a^{8} + 11 a^{7} + a^{6} + 14 a^{5} + 16 a^{4} + 9 a^{3} + 21 a + 18 + \left(21 a^{8} + 8 a^{7} + 8 a^{6} + a^{5} + 14 a^{4} + 9 a^{2} + 21 a + 5\right)\cdot 23 + \left(a^{8} + 18 a^{7} + 9 a^{5} + 2 a^{4} + 19 a^{3}\right)\cdot 23^{2} + \left(19 a^{8} + 19 a^{7} + 8 a^{6} + 10 a^{5} + 17 a^{4} + 2 a^{3} + 14 a^{2} + 4 a + 21\right)\cdot 23^{3} + \left(13 a^{8} + 10 a^{7} + 13 a^{6} + 12 a^{5} + 6 a^{4} + 14 a^{3} + 17 a^{2} + 16 a + 20\right)\cdot 23^{4} + \left(10 a^{8} + 18 a^{7} + 6 a^{5} + 14 a^{4} + a^{3} + 2 a^{2} + 6 a + 16\right)\cdot 23^{5} + \left(6 a^{8} + 2 a^{7} + 20 a^{6} + 2 a^{5} + 21 a^{4} + 4 a^{3} + 13 a^{2} + 13 a + 19\right)\cdot 23^{6} + \left(a^{8} + 12 a^{7} + a^{6} + 4 a^{5} + 21 a^{4} + 13 a^{3} + 6 a^{2} + 15 a + 17\right)\cdot 23^{7} +O(23^{8})$$ 19*a^8 + 11*a^7 + a^6 + 14*a^5 + 16*a^4 + 9*a^3 + 21*a + 18 + (21*a^8 + 8*a^7 + 8*a^6 + a^5 + 14*a^4 + 9*a^2 + 21*a + 5)*23 + (a^8 + 18*a^7 + 9*a^5 + 2*a^4 + 19*a^3)*23^2 + (19*a^8 + 19*a^7 + 8*a^6 + 10*a^5 + 17*a^4 + 2*a^3 + 14*a^2 + 4*a + 21)*23^3 + (13*a^8 + 10*a^7 + 13*a^6 + 12*a^5 + 6*a^4 + 14*a^3 + 17*a^2 + 16*a + 20)*23^4 + (10*a^8 + 18*a^7 + 6*a^5 + 14*a^4 + a^3 + 2*a^2 + 6*a + 16)*23^5 + (6*a^8 + 2*a^7 + 20*a^6 + 2*a^5 + 21*a^4 + 4*a^3 + 13*a^2 + 13*a + 19)*23^6 + (a^8 + 12*a^7 + a^6 + 4*a^5 + 21*a^4 + 13*a^3 + 6*a^2 + 15*a + 17)*23^7+O(23^8) $r_{ 15 }$ $=$ $$7 a^{8} + 20 a^{7} + a^{6} + 21 a^{5} + 5 a^{4} + 16 a^{3} + 16 a^{2} + 16 a + 1 + \left(17 a^{8} + 14 a^{7} + 6 a^{6} + 15 a^{5} + 11 a^{4} + 18 a^{3} + 15 a^{2} + 18 a + 13\right)\cdot 23 + \left(a^{8} + 7 a^{7} + 21 a^{6} + 16 a^{5} + 13 a^{4} + 21 a^{3} + 16 a^{2} + 22 a + 9\right)\cdot 23^{2} + \left(18 a^{8} + 19 a^{6} + 3 a^{5} + 18 a^{4} + 3 a^{3} + 20 a^{2} + 20 a + 20\right)\cdot 23^{3} + \left(7 a^{7} + 11 a^{6} + 16 a^{5} + 15 a^{4} + 7 a^{3} + 5 a^{2} + 3 a + 14\right)\cdot 23^{4} + \left(20 a^{8} + 19 a^{7} + 11 a^{6} + 5 a^{5} + 17 a^{4} + 4 a^{3} + 9 a^{2} + 9 a + 14\right)\cdot 23^{5} + \left(7 a^{8} + 17 a^{7} + 14 a^{6} + 11 a^{5} + 8 a^{4} + 17 a^{3} + 4 a^{2} + 4 a + 3\right)\cdot 23^{6} + \left(7 a^{8} + 17 a^{7} + 8 a^{6} + 21 a^{5} + 9 a^{4} + 19 a^{3} + 3 a^{2} + a + 5\right)\cdot 23^{7} +O(23^{8})$$ 7*a^8 + 20*a^7 + a^6 + 21*a^5 + 5*a^4 + 16*a^3 + 16*a^2 + 16*a + 1 + (17*a^8 + 14*a^7 + 6*a^6 + 15*a^5 + 11*a^4 + 18*a^3 + 15*a^2 + 18*a + 13)*23 + (a^8 + 7*a^7 + 21*a^6 + 16*a^5 + 13*a^4 + 21*a^3 + 16*a^2 + 22*a + 9)*23^2 + (18*a^8 + 19*a^6 + 3*a^5 + 18*a^4 + 3*a^3 + 20*a^2 + 20*a + 20)*23^3 + (7*a^7 + 11*a^6 + 16*a^5 + 15*a^4 + 7*a^3 + 5*a^2 + 3*a + 14)*23^4 + (20*a^8 + 19*a^7 + 11*a^6 + 5*a^5 + 17*a^4 + 4*a^3 + 9*a^2 + 9*a + 14)*23^5 + (7*a^8 + 17*a^7 + 14*a^6 + 11*a^5 + 8*a^4 + 17*a^3 + 4*a^2 + 4*a + 3)*23^6 + (7*a^8 + 17*a^7 + 8*a^6 + 21*a^5 + 9*a^4 + 19*a^3 + 3*a^2 + a + 5)*23^7+O(23^8) $r_{ 16 }$ $=$ $$6 a^{8} + 13 a^{7} + 18 a^{6} + 11 a^{5} + 3 a^{4} + 13 a^{3} + 11 a^{2} + 8 a + 9 + \left(10 a^{8} + 16 a^{7} + 16 a^{6} + 22 a^{5} + 19 a^{4} + 12 a^{3} + a^{2} + 22 a + 19\right)\cdot 23 + \left(2 a^{8} + 14 a^{7} + 19 a^{6} + 14 a^{4} + 9 a^{3} + 10 a^{2} + 15 a + 22\right)\cdot 23^{2} + \left(5 a^{7} + 18 a^{6} + 17 a^{4} + 17 a^{3} + 12 a^{2} + 5 a + 19\right)\cdot 23^{3} + \left(15 a^{8} + 7 a^{7} + 20 a^{6} + 10 a^{5} + 12 a^{4} + 21 a^{3} + 22 a^{2} + 2 a + 4\right)\cdot 23^{4} + \left(6 a^{8} + 21 a^{7} + 9 a^{6} + 21 a^{5} + 19 a^{4} + 22 a^{3} + 8 a^{2} + 11 a + 1\right)\cdot 23^{5} + \left(11 a^{8} + 4 a^{7} + 19 a^{6} + 2 a^{5} + 18 a^{4} + 7 a^{3} + 5 a^{2} + 13 a + 9\right)\cdot 23^{6} + \left(21 a^{8} + 20 a^{7} + 16 a^{6} + 22 a^{4} + 19 a^{3} + 22 a^{2} + 9 a + 9\right)\cdot 23^{7} +O(23^{8})$$ 6*a^8 + 13*a^7 + 18*a^6 + 11*a^5 + 3*a^4 + 13*a^3 + 11*a^2 + 8*a + 9 + (10*a^8 + 16*a^7 + 16*a^6 + 22*a^5 + 19*a^4 + 12*a^3 + a^2 + 22*a + 19)*23 + (2*a^8 + 14*a^7 + 19*a^6 + 14*a^4 + 9*a^3 + 10*a^2 + 15*a + 22)*23^2 + (5*a^7 + 18*a^6 + 17*a^4 + 17*a^3 + 12*a^2 + 5*a + 19)*23^3 + (15*a^8 + 7*a^7 + 20*a^6 + 10*a^5 + 12*a^4 + 21*a^3 + 22*a^2 + 2*a + 4)*23^4 + (6*a^8 + 21*a^7 + 9*a^6 + 21*a^5 + 19*a^4 + 22*a^3 + 8*a^2 + 11*a + 1)*23^5 + (11*a^8 + 4*a^7 + 19*a^6 + 2*a^5 + 18*a^4 + 7*a^3 + 5*a^2 + 13*a + 9)*23^6 + (21*a^8 + 20*a^7 + 16*a^6 + 22*a^4 + 19*a^3 + 22*a^2 + 9*a + 9)*23^7+O(23^8) $r_{ 17 }$ $=$ $$22 a^{8} + 15 a^{7} + 6 a^{6} + 12 a^{5} + 2 a^{4} + 14 a^{3} + 8 a^{2} + a + 13 + \left(8 a^{8} + 4 a^{7} + 18 a^{6} + a^{5} + 16 a^{4} + 3 a^{3} + 20 a^{2} + 19 a + 19\right)\cdot 23 + \left(18 a^{8} + 11 a^{7} + 22 a^{4} + 19 a^{3} + 19 a^{2} + 8 a + 8\right)\cdot 23^{2} + \left(14 a^{8} + 21 a^{7} + 9 a^{6} + 10 a^{5} + 7 a^{4} + 21 a^{3} + 15 a^{2} + 16 a + 4\right)\cdot 23^{3} + \left(13 a^{8} + 11 a^{7} + 2 a^{6} + 16 a^{5} + 22 a^{4} + 19 a^{3} + a^{2} + a + 4\right)\cdot 23^{4} + \left(10 a^{8} + 13 a^{7} + 19 a^{6} + 19 a^{5} + 15 a^{4} + 5 a^{3} + 4 a^{2} + 15 a + 5\right)\cdot 23^{5} + \left(11 a^{8} + 3 a^{7} + 5 a^{6} + 11 a^{5} + 20 a^{4} + 3 a^{3} + 16 a^{2} + 18 a + 13\right)\cdot 23^{6} + \left(19 a^{8} + 14 a^{7} + 2 a^{6} + 16 a^{5} + 19 a^{4} + 20 a^{3} + 6 a^{2} + 18\right)\cdot 23^{7} +O(23^{8})$$ 22*a^8 + 15*a^7 + 6*a^6 + 12*a^5 + 2*a^4 + 14*a^3 + 8*a^2 + a + 13 + (8*a^8 + 4*a^7 + 18*a^6 + a^5 + 16*a^4 + 3*a^3 + 20*a^2 + 19*a + 19)*23 + (18*a^8 + 11*a^7 + 22*a^4 + 19*a^3 + 19*a^2 + 8*a + 8)*23^2 + (14*a^8 + 21*a^7 + 9*a^6 + 10*a^5 + 7*a^4 + 21*a^3 + 15*a^2 + 16*a + 4)*23^3 + (13*a^8 + 11*a^7 + 2*a^6 + 16*a^5 + 22*a^4 + 19*a^3 + a^2 + a + 4)*23^4 + (10*a^8 + 13*a^7 + 19*a^6 + 19*a^5 + 15*a^4 + 5*a^3 + 4*a^2 + 15*a + 5)*23^5 + (11*a^8 + 3*a^7 + 5*a^6 + 11*a^5 + 20*a^4 + 3*a^3 + 16*a^2 + 18*a + 13)*23^6 + (19*a^8 + 14*a^7 + 2*a^6 + 16*a^5 + 19*a^4 + 20*a^3 + 6*a^2 + 18)*23^7+O(23^8) $r_{ 18 }$ $=$ $$12 a^{8} + 19 a^{7} + 17 a^{6} + 10 a^{5} + 2 a^{4} + 18 a^{3} + 10 a^{2} + 6 a + 8 + \left(7 a^{8} + 11 a^{7} + 16 a^{6} + 6 a^{5} + 8 a^{4} + 21 a^{3} + 22 a^{2} + 20 a + 11\right)\cdot 23 + \left(3 a^{8} + 10 a^{7} + 8 a^{6} + 20 a^{5} + 14 a^{4} + 6 a^{3} + 11 a^{2} + 10 a + 17\right)\cdot 23^{2} + \left(22 a^{8} + 15 a^{7} + 18 a^{6} + 11 a^{5} + 10 a^{4} + 7 a^{3} + 6 a^{2} + 9 a + 18\right)\cdot 23^{3} + \left(10 a^{8} + 4 a^{7} + 19 a^{6} + 13 a^{5} + 15 a^{4} + 9 a^{3} + 7 a^{2} + 11 a + 15\right)\cdot 23^{4} + \left(7 a^{8} + 16 a^{6} + 11 a^{5} + 3 a^{4} + 10 a^{3} + 14 a^{2} + 15 a + 20\right)\cdot 23^{5} + \left(12 a^{8} + 21 a^{7} + 19 a^{6} + 18 a^{5} + 17 a^{4} + 10 a^{3} + 15 a^{2} + a + 20\right)\cdot 23^{6} + \left(17 a^{8} + 7 a^{7} + 12 a^{6} + 6 a^{5} + 5 a^{4} + 5 a^{3} + 20 a^{2} + 10 a + 15\right)\cdot 23^{7} +O(23^{8})$$ 12*a^8 + 19*a^7 + 17*a^6 + 10*a^5 + 2*a^4 + 18*a^3 + 10*a^2 + 6*a + 8 + (7*a^8 + 11*a^7 + 16*a^6 + 6*a^5 + 8*a^4 + 21*a^3 + 22*a^2 + 20*a + 11)*23 + (3*a^8 + 10*a^7 + 8*a^6 + 20*a^5 + 14*a^4 + 6*a^3 + 11*a^2 + 10*a + 17)*23^2 + (22*a^8 + 15*a^7 + 18*a^6 + 11*a^5 + 10*a^4 + 7*a^3 + 6*a^2 + 9*a + 18)*23^3 + (10*a^8 + 4*a^7 + 19*a^6 + 13*a^5 + 15*a^4 + 9*a^3 + 7*a^2 + 11*a + 15)*23^4 + (7*a^8 + 16*a^6 + 11*a^5 + 3*a^4 + 10*a^3 + 14*a^2 + 15*a + 20)*23^5 + (12*a^8 + 21*a^7 + 19*a^6 + 18*a^5 + 17*a^4 + 10*a^3 + 15*a^2 + a + 20)*23^6 + (17*a^8 + 7*a^7 + 12*a^6 + 6*a^5 + 5*a^4 + 5*a^3 + 20*a^2 + 10*a + 15)*23^7+O(23^8)

Generators of the action on the roots $r_1, \ldots, r_{ 18 }$

 Cycle notation $(1,2)(3,11)(4,12)(5,18)(6,13)(7,17)(8,10)(9,15)(14,16)$ $(1,3,7,9,13,8,14,18,4)(2,11,17,15,6,10,16,5,12)$ $(1,15,14,2,9,16)(3,6,18,11,13,5)(4,17,8,12,7,10)$

Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 18 }$ Character value $1$ $1$ $()$ $1$ $1$ $2$ $(1,2)(3,11)(4,12)(5,18)(6,13)(7,17)(8,10)(9,15)(14,16)$ $-1$ $1$ $3$ $(1,9,14)(2,15,16)(3,13,18)(4,7,8)(5,11,6)(10,12,17)$ $-\zeta_{9}^{3} - 1$ $1$ $3$ $(1,14,9)(2,16,15)(3,18,13)(4,8,7)(5,6,11)(10,17,12)$ $\zeta_{9}^{3}$ $1$ $6$ $(1,15,14,2,9,16)(3,6,18,11,13,5)(4,17,8,12,7,10)$ $\zeta_{9}^{3} + 1$ $1$ $6$ $(1,16,9,2,14,15)(3,5,13,11,18,6)(4,10,7,12,8,17)$ $-\zeta_{9}^{3}$ $1$ $9$ $(1,3,7,9,13,8,14,18,4)(2,11,17,15,6,10,16,5,12)$ $\zeta_{9}^{2}$ $1$ $9$ $(1,7,13,14,4,3,9,8,18)(2,17,6,16,12,11,15,10,5)$ $\zeta_{9}^{4}$ $1$ $9$ $(1,13,4,9,18,7,14,3,8)(2,6,12,15,5,17,16,11,10)$ $-\zeta_{9}^{5} - \zeta_{9}^{2}$ $1$ $9$ $(1,8,3,14,7,18,9,4,13)(2,10,11,16,17,5,15,12,6)$ $\zeta_{9}$ $1$ $9$ $(1,18,8,9,3,4,14,13,7)(2,5,10,15,11,12,16,6,17)$ $\zeta_{9}^{5}$ $1$ $9$ $(1,4,18,14,8,13,9,7,3)(2,12,5,16,10,6,15,17,11)$ $-\zeta_{9}^{4} - \zeta_{9}$ $1$ $18$ $(1,11,7,15,13,10,14,5,4,2,3,17,9,6,8,16,18,12)$ $-\zeta_{9}^{2}$ $1$ $18$ $(1,10,3,16,7,5,9,12,13,2,8,11,14,17,18,15,4,6)$ $-\zeta_{9}$ $1$ $18$ $(1,5,8,15,3,12,14,6,7,2,18,10,9,11,4,16,13,17)$ $-\zeta_{9}^{5}$ $1$ $18$ $(1,17,13,16,4,11,9,10,18,2,7,6,14,12,3,15,8,5)$ $-\zeta_{9}^{4}$ $1$ $18$ $(1,6,4,15,18,17,14,11,8,2,13,12,9,5,7,16,3,10)$ $\zeta_{9}^{5} + \zeta_{9}^{2}$ $1$ $18$ $(1,12,18,16,8,6,9,17,3,2,4,5,14,10,13,15,7,11)$ $\zeta_{9}^{4} + \zeta_{9}$

The blue line marks the conjugacy class containing complex conjugation.