# Properties

 Label 2.729.18t16.a.c Dimension $2$ Group $C_9\times S_3$ Conductor $729$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $C_9\times S_3$ Conductor: $$729$$$$\medspace = 3^{6}$$ Artin stem field: Galois closure of 18.0.2954312706550833698643.2 Galois orbit size: $6$ Smallest permutation container: $C_9\times S_3$ Parity: odd Determinant: 1.27.18t1.a.d Projective image: $S_3$ Projective stem field: Galois closure of 3.1.243.1

## Defining polynomial

 $f(x)$ $=$ $$x^{18} - 9x^{15} + 36x^{12} - 53x^{9} + 27x^{6} + 1$$ x^18 - 9*x^15 + 36*x^12 - 53*x^9 + 27*x^6 + 1 .

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $$x^{9} + 4x^{3} + 20x^{2} + 29x + 28$$

Roots:
 $r_{ 1 }$ $=$ $$30 a^{8} + 3 a^{7} + 24 a^{6} + 19 a^{5} + 9 a^{4} + 8 a^{3} + 14 a^{2} + 2 a + 16 + \left(6 a^{7} + 15 a^{6} + 11 a^{5} + 25 a^{4} + 5 a^{3} + 22 a + 27\right)\cdot 31 + \left(30 a^{8} + 9 a^{7} + 22 a^{6} + 29 a^{5} + 19 a^{4} + 23 a^{3} + 20 a^{2} + 19 a + 5\right)\cdot 31^{2} + \left(30 a^{8} + 23 a^{7} + 18 a^{5} + 28 a^{3} + 15 a^{2} + 21 a + 12\right)\cdot 31^{3} + \left(4 a^{8} + 12 a^{7} + 29 a^{6} + 12 a^{5} + 29 a^{4} + 3 a^{3} + 10 a^{2} + 28 a + 15\right)\cdot 31^{4} + \left(22 a^{8} + 13 a^{7} + 13 a^{6} + 3 a^{5} + 19 a^{3} + 24 a^{2} + 14\right)\cdot 31^{5} + \left(16 a^{8} + 13 a^{7} + 4 a^{6} + 3 a^{5} + a^{4} + 13 a^{3} + 25 a^{2} + 4 a + 14\right)\cdot 31^{6} + \left(5 a^{8} + 25 a^{7} + 12 a^{6} + 29 a^{5} + 24 a^{4} + 27 a^{3} + 17 a^{2} + 4 a + 2\right)\cdot 31^{7} + \left(9 a^{8} + 15 a^{7} + 20 a^{6} + 18 a^{5} + 22 a^{4} + 3 a^{3} + 30 a^{2} + 2 a + 10\right)\cdot 31^{8} + \left(4 a^{8} + 15 a^{7} + 2 a^{6} + 5 a^{5} + 12 a^{4} + 24 a^{3} + 6 a^{2} + 25 a + 4\right)\cdot 31^{9} +O(31^{10})$$ 30*a^8 + 3*a^7 + 24*a^6 + 19*a^5 + 9*a^4 + 8*a^3 + 14*a^2 + 2*a + 16 + (6*a^7 + 15*a^6 + 11*a^5 + 25*a^4 + 5*a^3 + 22*a + 27)*31 + (30*a^8 + 9*a^7 + 22*a^6 + 29*a^5 + 19*a^4 + 23*a^3 + 20*a^2 + 19*a + 5)*31^2 + (30*a^8 + 23*a^7 + 18*a^5 + 28*a^3 + 15*a^2 + 21*a + 12)*31^3 + (4*a^8 + 12*a^7 + 29*a^6 + 12*a^5 + 29*a^4 + 3*a^3 + 10*a^2 + 28*a + 15)*31^4 + (22*a^8 + 13*a^7 + 13*a^6 + 3*a^5 + 19*a^3 + 24*a^2 + 14)*31^5 + (16*a^8 + 13*a^7 + 4*a^6 + 3*a^5 + a^4 + 13*a^3 + 25*a^2 + 4*a + 14)*31^6 + (5*a^8 + 25*a^7 + 12*a^6 + 29*a^5 + 24*a^4 + 27*a^3 + 17*a^2 + 4*a + 2)*31^7 + (9*a^8 + 15*a^7 + 20*a^6 + 18*a^5 + 22*a^4 + 3*a^3 + 30*a^2 + 2*a + 10)*31^8 + (4*a^8 + 15*a^7 + 2*a^6 + 5*a^5 + 12*a^4 + 24*a^3 + 6*a^2 + 25*a + 4)*31^9+O(31^10) $r_{ 2 }$ $=$ $$9 a^{8} + 29 a^{7} + 30 a^{6} + 28 a^{5} + 4 a^{4} + 29 a^{3} + 29 a^{2} + 17 a + 26 + \left(11 a^{8} + 15 a^{7} + 29 a^{6} + 28 a^{5} + 19 a^{4} + 19 a^{3} + 22 a + 29\right)\cdot 31 + \left(11 a^{8} + 29 a^{7} + 21 a^{6} + 14 a^{5} + 11 a^{4} + 30 a^{3} + 19 a^{2} + 25 a + 14\right)\cdot 31^{2} + \left(22 a^{7} + 26 a^{6} + 23 a^{5} + 18 a^{4} + 6 a^{3} + 29 a^{2} + 27 a + 17\right)\cdot 31^{3} + \left(11 a^{8} + 23 a^{7} + 12 a^{6} + 25 a^{5} + 16 a^{4} + 22 a^{3} + 29\right)\cdot 31^{4} + \left(8 a^{8} + 14 a^{7} + 13 a^{6} + 14 a^{5} + 12 a^{4} + 4 a^{3} + 16 a^{2} + 18 a + 8\right)\cdot 31^{5} + \left(29 a^{8} + 24 a^{7} + 23 a^{6} + 29 a^{5} + 21 a^{4} + 27 a^{3} + 22 a^{2} + 9 a + 16\right)\cdot 31^{6} + \left(15 a^{8} + 17 a^{7} + 19 a^{6} + 9 a^{5} + 28 a^{4} + 9 a^{3} + 2 a^{2} + 16 a + 20\right)\cdot 31^{7} + \left(15 a^{8} + 10 a^{7} + 7 a^{6} + 6 a^{5} + 9 a^{4} + 25 a^{2} + 2 a + 2\right)\cdot 31^{8} + \left(5 a^{8} + 11 a^{7} + 30 a^{6} + 16 a^{5} + 7 a^{4} + 3 a^{3} + 7 a^{2} + 28 a + 13\right)\cdot 31^{9} +O(31^{10})$$ 9*a^8 + 29*a^7 + 30*a^6 + 28*a^5 + 4*a^4 + 29*a^3 + 29*a^2 + 17*a + 26 + (11*a^8 + 15*a^7 + 29*a^6 + 28*a^5 + 19*a^4 + 19*a^3 + 22*a + 29)*31 + (11*a^8 + 29*a^7 + 21*a^6 + 14*a^5 + 11*a^4 + 30*a^3 + 19*a^2 + 25*a + 14)*31^2 + (22*a^7 + 26*a^6 + 23*a^5 + 18*a^4 + 6*a^3 + 29*a^2 + 27*a + 17)*31^3 + (11*a^8 + 23*a^7 + 12*a^6 + 25*a^5 + 16*a^4 + 22*a^3 + 29)*31^4 + (8*a^8 + 14*a^7 + 13*a^6 + 14*a^5 + 12*a^4 + 4*a^3 + 16*a^2 + 18*a + 8)*31^5 + (29*a^8 + 24*a^7 + 23*a^6 + 29*a^5 + 21*a^4 + 27*a^3 + 22*a^2 + 9*a + 16)*31^6 + (15*a^8 + 17*a^7 + 19*a^6 + 9*a^5 + 28*a^4 + 9*a^3 + 2*a^2 + 16*a + 20)*31^7 + (15*a^8 + 10*a^7 + 7*a^6 + 6*a^5 + 9*a^4 + 25*a^2 + 2*a + 2)*31^8 + (5*a^8 + 11*a^7 + 30*a^6 + 16*a^5 + 7*a^4 + 3*a^3 + 7*a^2 + 28*a + 13)*31^9+O(31^10) $r_{ 3 }$ $=$ $$4 a^{8} + 30 a^{7} + 30 a^{6} + 24 a^{5} + 9 a^{4} + 11 a^{3} + 14 a^{2} + 29 a + 16 + \left(5 a^{8} + 18 a^{7} + 22 a^{6} + 14 a^{5} + 4 a^{4} + 30 a^{3} + 8 a^{2} + 30 a + 23\right)\cdot 31 + \left(30 a^{8} + 8 a^{7} + 20 a^{6} + 14 a^{5} + 23 a^{4} + 19 a^{3} + 9 a^{2} + 14 a + 12\right)\cdot 31^{2} + \left(29 a^{8} + 19 a^{7} + 28 a^{6} + 11 a^{5} + 21 a^{4} + 5 a^{3} + 22 a^{2} + 25 a + 5\right)\cdot 31^{3} + \left(3 a^{8} + 8 a^{7} + 7 a^{6} + 26 a^{5} + 16 a^{4} + 2 a^{3} + 20 a^{2} + 27 a + 30\right)\cdot 31^{4} + \left(4 a^{8} + 13 a^{7} + 23 a^{6} + 2 a^{5} + 9 a^{4} + 8 a^{3} + 24 a^{2} + 26 a + 2\right)\cdot 31^{5} + \left(12 a^{8} + 27 a^{7} + 3 a^{6} + 14 a^{5} + 26 a^{4} + 8 a^{3} + 14 a^{2} + 27 a + 26\right)\cdot 31^{6} + \left(17 a^{8} + 20 a^{7} + 15 a^{6} + 12 a^{5} + 16 a^{4} + 3 a^{3} + 4 a^{2} + 4 a + 17\right)\cdot 31^{7} + \left(19 a^{8} + 26 a^{7} + 7 a^{6} + 19 a^{5} + 21 a^{4} + 17 a^{3} + 10 a^{2} + 16 a + 12\right)\cdot 31^{8} + \left(21 a^{8} + 23 a^{7} + 14 a^{6} + 20 a^{5} + 17 a^{4} + 17 a^{3} + 9 a^{2} + 10 a + 26\right)\cdot 31^{9} +O(31^{10})$$ 4*a^8 + 30*a^7 + 30*a^6 + 24*a^5 + 9*a^4 + 11*a^3 + 14*a^2 + 29*a + 16 + (5*a^8 + 18*a^7 + 22*a^6 + 14*a^5 + 4*a^4 + 30*a^3 + 8*a^2 + 30*a + 23)*31 + (30*a^8 + 8*a^7 + 20*a^6 + 14*a^5 + 23*a^4 + 19*a^3 + 9*a^2 + 14*a + 12)*31^2 + (29*a^8 + 19*a^7 + 28*a^6 + 11*a^5 + 21*a^4 + 5*a^3 + 22*a^2 + 25*a + 5)*31^3 + (3*a^8 + 8*a^7 + 7*a^6 + 26*a^5 + 16*a^4 + 2*a^3 + 20*a^2 + 27*a + 30)*31^4 + (4*a^8 + 13*a^7 + 23*a^6 + 2*a^5 + 9*a^4 + 8*a^3 + 24*a^2 + 26*a + 2)*31^5 + (12*a^8 + 27*a^7 + 3*a^6 + 14*a^5 + 26*a^4 + 8*a^3 + 14*a^2 + 27*a + 26)*31^6 + (17*a^8 + 20*a^7 + 15*a^6 + 12*a^5 + 16*a^4 + 3*a^3 + 4*a^2 + 4*a + 17)*31^7 + (19*a^8 + 26*a^7 + 7*a^6 + 19*a^5 + 21*a^4 + 17*a^3 + 10*a^2 + 16*a + 12)*31^8 + (21*a^8 + 23*a^7 + 14*a^6 + 20*a^5 + 17*a^4 + 17*a^3 + 9*a^2 + 10*a + 26)*31^9+O(31^10) $r_{ 4 }$ $=$ $$7 a^{8} + 14 a^{7} + 12 a^{6} + 22 a^{5} + 16 a^{4} + 29 a^{2} + 21 a + 10 + \left(28 a^{8} + 3 a^{7} + 4 a^{6} + 17 a^{5} + 5 a^{4} + 3 a^{3} + 10 a^{2} + 4 a + 28\right)\cdot 31 + \left(14 a^{8} + 28 a^{7} + 20 a^{6} + 17 a^{5} + 15 a^{4} + 14 a^{3} + 11 a^{2} + 16 a + 3\right)\cdot 31^{2} + \left(12 a^{8} + 23 a^{7} + 20 a^{6} + 23 a^{5} + 18 a^{4} + a^{3} + 17 a^{2} + 17 a + 8\right)\cdot 31^{3} + \left(30 a^{8} + 22 a^{7} + 29 a^{6} + 4 a^{4} + 21 a^{3} + 14 a^{2} + 6 a + 25\right)\cdot 31^{4} + \left(9 a^{8} + 10 a^{7} + 13 a^{6} + 14 a^{5} + 25 a^{4} + 24 a^{3} + 23 a^{2} + 13 a + 6\right)\cdot 31^{5} + \left(6 a^{8} + 2 a^{7} + 19 a^{6} + a^{5} + 3 a^{4} + 7 a^{3} + 5 a^{2} + 11 a + 10\right)\cdot 31^{6} + \left(4 a^{8} + 9 a^{7} + 28 a^{6} + 24 a^{5} + 23 a^{4} + 16 a^{3} + 9 a^{2} + 25 a + 19\right)\cdot 31^{7} + \left(17 a^{8} + 19 a^{7} + 4 a^{6} + 19 a^{5} + 12 a^{4} + 13 a^{3} + 20 a^{2} + 22 a\right)\cdot 31^{8} + \left(27 a^{8} + 5 a^{7} + 17 a^{6} + 3 a^{5} + 10 a^{4} + 7 a^{3} + 2 a^{2} + 10 a + 3\right)\cdot 31^{9} +O(31^{10})$$ 7*a^8 + 14*a^7 + 12*a^6 + 22*a^5 + 16*a^4 + 29*a^2 + 21*a + 10 + (28*a^8 + 3*a^7 + 4*a^6 + 17*a^5 + 5*a^4 + 3*a^3 + 10*a^2 + 4*a + 28)*31 + (14*a^8 + 28*a^7 + 20*a^6 + 17*a^5 + 15*a^4 + 14*a^3 + 11*a^2 + 16*a + 3)*31^2 + (12*a^8 + 23*a^7 + 20*a^6 + 23*a^5 + 18*a^4 + a^3 + 17*a^2 + 17*a + 8)*31^3 + (30*a^8 + 22*a^7 + 29*a^6 + 4*a^4 + 21*a^3 + 14*a^2 + 6*a + 25)*31^4 + (9*a^8 + 10*a^7 + 13*a^6 + 14*a^5 + 25*a^4 + 24*a^3 + 23*a^2 + 13*a + 6)*31^5 + (6*a^8 + 2*a^7 + 19*a^6 + a^5 + 3*a^4 + 7*a^3 + 5*a^2 + 11*a + 10)*31^6 + (4*a^8 + 9*a^7 + 28*a^6 + 24*a^5 + 23*a^4 + 16*a^3 + 9*a^2 + 25*a + 19)*31^7 + (17*a^8 + 19*a^7 + 4*a^6 + 19*a^5 + 12*a^4 + 13*a^3 + 20*a^2 + 22*a)*31^8 + (27*a^8 + 5*a^7 + 17*a^6 + 3*a^5 + 10*a^4 + 7*a^3 + 2*a^2 + 10*a + 3)*31^9+O(31^10) $r_{ 5 }$ $=$ $$24 a^{8} + 14 a^{7} + 17 a^{6} + 6 a^{5} + 12 a^{4} + 9 a^{3} + 17 a^{2} + 18 a + \left(a^{8} + 10 a^{7} + 27 a^{6} + 15 a^{5} + 11 a^{4} + 15 a^{3} + 20 a + 20\right)\cdot 31 + \left(20 a^{8} + 26 a^{7} + 28 a^{6} + 20 a^{5} + 8 a^{4} + 24 a^{3} + 21 a^{2} + 9 a + 18\right)\cdot 31^{2} + \left(29 a^{8} + 15 a^{7} + 15 a^{6} + 30 a^{5} + 4 a^{4} + 24 a^{3} + 8 a^{2} + 15 a + 9\right)\cdot 31^{3} + \left(23 a^{8} + 24 a^{7} + 9 a^{6} + 9 a^{5} + 24 a^{4} + 25 a^{3} + 28 a^{2} + 25\right)\cdot 31^{4} + \left(11 a^{8} + 16 a^{7} + 26 a^{6} + 5 a^{5} + 23 a^{4} + 11 a^{3} + 19 a^{2} + 30 a + 14\right)\cdot 31^{5} + \left(8 a^{8} + 22 a^{7} + 5 a^{6} + 10 a^{5} + 6 a^{4} + 3 a^{3} + 17 a^{2} + 24 a + 10\right)\cdot 31^{6} + \left(20 a^{8} + 4 a^{7} + 17 a^{6} + 7 a^{5} + 22 a^{4} + 3 a^{3} + 20 a^{2} + 12 a + 25\right)\cdot 31^{7} + \left(6 a^{8} + 27 a^{7} + 25 a^{5} + 12 a^{4} + 7 a^{3} + a^{2} + 12 a + 9\right)\cdot 31^{8} + \left(6 a^{8} + a^{7} + 9 a^{6} + 14 a^{5} + 24 a^{4} + 22 a^{3} + 6 a^{2} + 3\right)\cdot 31^{9} +O(31^{10})$$ 24*a^8 + 14*a^7 + 17*a^6 + 6*a^5 + 12*a^4 + 9*a^3 + 17*a^2 + 18*a + (a^8 + 10*a^7 + 27*a^6 + 15*a^5 + 11*a^4 + 15*a^3 + 20*a + 20)*31 + (20*a^8 + 26*a^7 + 28*a^6 + 20*a^5 + 8*a^4 + 24*a^3 + 21*a^2 + 9*a + 18)*31^2 + (29*a^8 + 15*a^7 + 15*a^6 + 30*a^5 + 4*a^4 + 24*a^3 + 8*a^2 + 15*a + 9)*31^3 + (23*a^8 + 24*a^7 + 9*a^6 + 9*a^5 + 24*a^4 + 25*a^3 + 28*a^2 + 25)*31^4 + (11*a^8 + 16*a^7 + 26*a^6 + 5*a^5 + 23*a^4 + 11*a^3 + 19*a^2 + 30*a + 14)*31^5 + (8*a^8 + 22*a^7 + 5*a^6 + 10*a^5 + 6*a^4 + 3*a^3 + 17*a^2 + 24*a + 10)*31^6 + (20*a^8 + 4*a^7 + 17*a^6 + 7*a^5 + 22*a^4 + 3*a^3 + 20*a^2 + 12*a + 25)*31^7 + (6*a^8 + 27*a^7 + 25*a^5 + 12*a^4 + 7*a^3 + a^2 + 12*a + 9)*31^8 + (6*a^8 + a^7 + 9*a^6 + 14*a^5 + 24*a^4 + 22*a^3 + 6*a^2 + 3)*31^9+O(31^10) $r_{ 6 }$ $=$ $$28 a^{8} + 5 a^{7} + 26 a^{6} + 18 a^{5} + 5 a^{4} + 26 a^{3} + 7 a^{2} + 6 a + 25 + \left(10 a^{8} + 26 a^{7} + 13 a^{6} + 12 a^{5} + 5 a^{4} + 5 a^{3} + 22 a^{2} + 6 a + 18\right)\cdot 31 + \left(26 a^{8} + 15 a^{7} + 2 a^{6} + 24 a^{5} + a^{4} + 2 a^{3} + 13 a^{2} + 26 a + 12\right)\cdot 31^{2} + \left(5 a^{8} + 26 a^{7} + 18 a^{6} + 5 a^{5} + 28 a^{4} + 4 a^{3} + 27 a^{2} + 13 a + 22\right)\cdot 31^{3} + \left(a^{8} + 2 a^{7} + 15 a^{6} + 9 a^{5} + 23 a^{4} + 5 a^{3} + 18 a^{2} + 2 a + 16\right)\cdot 31^{4} + \left(22 a^{7} + a^{6} + 25 a^{5} + 22 a^{4} + 5 a^{3} + 9 a^{2} + 28 a + 14\right)\cdot 31^{5} + \left(24 a^{8} + 18 a^{7} + 16 a^{6} + 24 a^{5} + 8 a^{4} + 23 a^{3} + 19 a^{2} + 2 a + 29\right)\cdot 31^{6} + \left(7 a^{8} + 29 a^{7} + 14 a^{6} + 12 a^{5} + 15 a^{4} + 2 a^{3} + 14 a^{2} + 23 a + 27\right)\cdot 31^{7} + \left(23 a^{8} + 7 a^{7} + 2 a^{6} + 27 a^{5} + 19 a^{4} + 9 a^{3} + 25 a^{2} + 9 a + 2\right)\cdot 31^{8} + \left(2 a^{8} + 7 a^{7} + 4 a^{6} + 16 a^{5} + 14 a^{4} + 2 a^{3} + 30 a^{2} + 19 a + 12\right)\cdot 31^{9} +O(31^{10})$$ 28*a^8 + 5*a^7 + 26*a^6 + 18*a^5 + 5*a^4 + 26*a^3 + 7*a^2 + 6*a + 25 + (10*a^8 + 26*a^7 + 13*a^6 + 12*a^5 + 5*a^4 + 5*a^3 + 22*a^2 + 6*a + 18)*31 + (26*a^8 + 15*a^7 + 2*a^6 + 24*a^5 + a^4 + 2*a^3 + 13*a^2 + 26*a + 12)*31^2 + (5*a^8 + 26*a^7 + 18*a^6 + 5*a^5 + 28*a^4 + 4*a^3 + 27*a^2 + 13*a + 22)*31^3 + (a^8 + 2*a^7 + 15*a^6 + 9*a^5 + 23*a^4 + 5*a^3 + 18*a^2 + 2*a + 16)*31^4 + (22*a^7 + a^6 + 25*a^5 + 22*a^4 + 5*a^3 + 9*a^2 + 28*a + 14)*31^5 + (24*a^8 + 18*a^7 + 16*a^6 + 24*a^5 + 8*a^4 + 23*a^3 + 19*a^2 + 2*a + 29)*31^6 + (7*a^8 + 29*a^7 + 14*a^6 + 12*a^5 + 15*a^4 + 2*a^3 + 14*a^2 + 23*a + 27)*31^7 + (23*a^8 + 7*a^7 + 2*a^6 + 27*a^5 + 19*a^4 + 9*a^3 + 25*a^2 + 9*a + 2)*31^8 + (2*a^8 + 7*a^7 + 4*a^6 + 16*a^5 + 14*a^4 + 2*a^3 + 30*a^2 + 19*a + 12)*31^9+O(31^10) $r_{ 7 }$ $=$ $$6 a^{8} + 13 a^{7} + 11 a^{6} + 10 a^{5} + 8 a^{4} + 14 a^{3} + 9 a^{2} + 19 a + 28 + \left(8 a^{8} + 14 a^{7} + 3 a^{6} + 5 a^{5} + a^{4} + 11 a^{3} + 18 a^{2} + 25 a + 13\right)\cdot 31 + \left(16 a^{8} + 4 a^{7} + 29 a^{6} + 11 a^{5} + 24 a^{4} + 29 a^{3} + 2 a^{2} + 20 a + 1\right)\cdot 31^{2} + \left(29 a^{8} + 26 a^{7} + 8 a^{6} + 5 a^{5} + 25 a^{3} + 30 a^{2} + 9 a + 17\right)\cdot 31^{3} + \left(3 a^{8} + 17 a^{7} + 19 a^{6} + 7 a^{5} + 9 a^{3} + 4 a^{2} + 17 a + 17\right)\cdot 31^{4} + \left(3 a^{8} + 27 a^{7} + 6 a^{6} + 27 a^{5} + 23 a^{3} + 28 a^{2} + 10 a + 10\right)\cdot 31^{5} + \left(13 a^{8} + 24 a^{7} + 14 a^{6} + 22 a^{5} + 23 a^{4} + 28 a^{3} + 18 a^{2} + 5 a + 15\right)\cdot 31^{6} + \left(12 a^{8} + 22 a^{7} + 26 a^{6} + 3 a^{5} + 7 a^{4} + 20 a^{3} + 22 a^{2} + 8 a + 1\right)\cdot 31^{7} + \left(29 a^{8} + 4 a^{7} + 9 a^{6} + 8 a^{5} + 18 a^{4} + 20 a^{3} + 11 a^{2} + 5 a + 7\right)\cdot 31^{8} + \left(13 a^{8} + 22 a^{6} + 13 a^{5} + a^{4} + 27 a^{3} + 18 a^{2} + 24 a + 1\right)\cdot 31^{9} +O(31^{10})$$ 6*a^8 + 13*a^7 + 11*a^6 + 10*a^5 + 8*a^4 + 14*a^3 + 9*a^2 + 19*a + 28 + (8*a^8 + 14*a^7 + 3*a^6 + 5*a^5 + a^4 + 11*a^3 + 18*a^2 + 25*a + 13)*31 + (16*a^8 + 4*a^7 + 29*a^6 + 11*a^5 + 24*a^4 + 29*a^3 + 2*a^2 + 20*a + 1)*31^2 + (29*a^8 + 26*a^7 + 8*a^6 + 5*a^5 + 25*a^3 + 30*a^2 + 9*a + 17)*31^3 + (3*a^8 + 17*a^7 + 19*a^6 + 7*a^5 + 9*a^3 + 4*a^2 + 17*a + 17)*31^4 + (3*a^8 + 27*a^7 + 6*a^6 + 27*a^5 + 23*a^3 + 28*a^2 + 10*a + 10)*31^5 + (13*a^8 + 24*a^7 + 14*a^6 + 22*a^5 + 23*a^4 + 28*a^3 + 18*a^2 + 5*a + 15)*31^6 + (12*a^8 + 22*a^7 + 26*a^6 + 3*a^5 + 7*a^4 + 20*a^3 + 22*a^2 + 8*a + 1)*31^7 + (29*a^8 + 4*a^7 + 9*a^6 + 8*a^5 + 18*a^4 + 20*a^3 + 11*a^2 + 5*a + 7)*31^8 + (13*a^8 + 22*a^6 + 13*a^5 + a^4 + 27*a^3 + 18*a^2 + 24*a + 1)*31^9+O(31^10) $r_{ 8 }$ $=$ $$8 a^{8} + 12 a^{7} + 6 a^{6} + 18 a^{5} + 7 a^{4} + 12 a^{3} + 12 a^{2} + 22 a + 30 + \left(23 a^{8} + 25 a^{7} + 20 a^{6} + 23 a^{5} + 15 a^{4} + a^{3} + 22 a^{2} + 29 a + 14\right)\cdot 31 + \left(22 a^{8} + 19 a^{7} + 30 a^{6} + 11 a^{5} + 19 a^{3} + 13 a^{2} + 20 a + 15\right)\cdot 31^{2} + \left(22 a^{8} + 12 a^{7} + 29 a^{6} + 3 a^{5} + 11 a^{4} + 27 a^{3} + 14 a^{2} + 4 a + 1\right)\cdot 31^{3} + \left(23 a^{8} + 7 a^{7} + 8 a^{6} + 4 a^{5} + 7 a^{4} + 25 a^{3} + 19 a^{2} + 20 a\right)\cdot 31^{4} + \left(6 a^{8} + 20 a^{7} + 24 a^{6} + 15 a^{5} + 29 a^{2} + 28 a + 27\right)\cdot 31^{5} + \left(29 a^{8} + 30 a^{7} + 19 a^{6} + 15 a^{5} + 15 a^{4} + 28 a^{3} + 25 a^{2} + 8 a + 4\right)\cdot 31^{6} + \left(6 a^{8} + 27 a^{7} + 3 a^{6} + a^{4} + 3 a^{3} + 18 a^{2} + 27 a + 28\right)\cdot 31^{7} + \left(11 a^{8} + 18 a^{7} + 3 a^{6} + 12 a^{5} + 25 a^{4} + 30 a^{3} + 22 a^{2} + 13 a + 19\right)\cdot 31^{8} + \left(25 a^{8} + 16 a^{6} + 22 a^{5} + 21 a^{4} + 7 a^{3} + 15 a^{2} + 18 a + 27\right)\cdot 31^{9} +O(31^{10})$$ 8*a^8 + 12*a^7 + 6*a^6 + 18*a^5 + 7*a^4 + 12*a^3 + 12*a^2 + 22*a + 30 + (23*a^8 + 25*a^7 + 20*a^6 + 23*a^5 + 15*a^4 + a^3 + 22*a^2 + 29*a + 14)*31 + (22*a^8 + 19*a^7 + 30*a^6 + 11*a^5 + 19*a^3 + 13*a^2 + 20*a + 15)*31^2 + (22*a^8 + 12*a^7 + 29*a^6 + 3*a^5 + 11*a^4 + 27*a^3 + 14*a^2 + 4*a + 1)*31^3 + (23*a^8 + 7*a^7 + 8*a^6 + 4*a^5 + 7*a^4 + 25*a^3 + 19*a^2 + 20*a)*31^4 + (6*a^8 + 20*a^7 + 24*a^6 + 15*a^5 + 29*a^2 + 28*a + 27)*31^5 + (29*a^8 + 30*a^7 + 19*a^6 + 15*a^5 + 15*a^4 + 28*a^3 + 25*a^2 + 8*a + 4)*31^6 + (6*a^8 + 27*a^7 + 3*a^6 + a^4 + 3*a^3 + 18*a^2 + 27*a + 28)*31^7 + (11*a^8 + 18*a^7 + 3*a^6 + 12*a^5 + 25*a^4 + 30*a^3 + 22*a^2 + 13*a + 19)*31^8 + (25*a^8 + 16*a^6 + 22*a^5 + 21*a^4 + 7*a^3 + 15*a^2 + 18*a + 27)*31^9+O(31^10) $r_{ 9 }$ $=$ $$7 a^{8} + 6 a^{7} + 6 a^{6} + 11 a^{5} + 8 a^{4} + 27 a^{3} + 9 a^{2} + 12 a + 28 + \left(6 a^{8} + 24 a^{7} + 9 a^{5} + 3 a^{4} + 4 a^{3} + a^{2} + 28 a + 6\right)\cdot 31 + \left(23 a^{8} + 16 a^{7} + 12 a^{6} + 29 a^{5} + 19 a^{4} + 28 a^{3} + 17 a^{2} + 20 a + 16\right)\cdot 31^{2} + \left(17 a^{8} + 29 a^{7} + 17 a^{6} + 18 a^{5} + 21 a^{4} + 16 a^{3} + 10 a^{2} + 26 a + 26\right)\cdot 31^{3} + \left(21 a^{8} + 19 a^{7} + 3 a^{6} + 23 a^{5} + 21 a^{4} + 10 a^{3} + 13 a^{2} + 20 a + 14\right)\cdot 31^{4} + \left(21 a^{8} + 9 a^{7} + 30 a^{6} + 27 a^{5} + 27 a^{4} + 6 a^{2} + 16 a + 21\right)\cdot 31^{5} + \left(5 a^{8} + 5 a^{7} + 17 a^{6} + 14 a^{5} + 23 a^{4} + a^{3} + 22 a^{2} + 13 a + 15\right)\cdot 31^{6} + \left(9 a^{8} + 21 a^{7} + 20 a^{6} + 26 a^{5} + 20 a^{4} + 5 a^{3} + 18 a^{2} + 14 a + 28\right)\cdot 31^{7} + \left(4 a^{8} + 4 a^{7} + 22 a^{6} + 19 a^{5} + 28 a^{4} + 26 a^{3} + 27 a^{2} + 21 a + 2\right)\cdot 31^{8} + \left(6 a^{8} + 4 a^{7} + a^{6} + 28 a^{5} + 19 a^{4} + a^{3} + 25 a^{2} + 29 a\right)\cdot 31^{9} +O(31^{10})$$ 7*a^8 + 6*a^7 + 6*a^6 + 11*a^5 + 8*a^4 + 27*a^3 + 9*a^2 + 12*a + 28 + (6*a^8 + 24*a^7 + 9*a^5 + 3*a^4 + 4*a^3 + a^2 + 28*a + 6)*31 + (23*a^8 + 16*a^7 + 12*a^6 + 29*a^5 + 19*a^4 + 28*a^3 + 17*a^2 + 20*a + 16)*31^2 + (17*a^8 + 29*a^7 + 17*a^6 + 18*a^5 + 21*a^4 + 16*a^3 + 10*a^2 + 26*a + 26)*31^3 + (21*a^8 + 19*a^7 + 3*a^6 + 23*a^5 + 21*a^4 + 10*a^3 + 13*a^2 + 20*a + 14)*31^4 + (21*a^8 + 9*a^7 + 30*a^6 + 27*a^5 + 27*a^4 + 6*a^2 + 16*a + 21)*31^5 + (5*a^8 + 5*a^7 + 17*a^6 + 14*a^5 + 23*a^4 + a^3 + 22*a^2 + 13*a + 15)*31^6 + (9*a^8 + 21*a^7 + 20*a^6 + 26*a^5 + 20*a^4 + 5*a^3 + 18*a^2 + 14*a + 28)*31^7 + (4*a^8 + 4*a^7 + 22*a^6 + 19*a^5 + 28*a^4 + 26*a^3 + 27*a^2 + 21*a + 2)*31^8 + (6*a^8 + 4*a^7 + a^6 + 28*a^5 + 19*a^4 + a^3 + 25*a^2 + 29*a)*31^9+O(31^10) $r_{ 10 }$ $=$ $$20 a^{8} + 9 a^{7} + 21 a^{6} + 23 a^{5} + 28 a^{4} + 12 a^{2} + 29 a + 2 + \left(11 a^{8} + 22 a^{6} + 19 a^{5} + 21 a^{4} + 13 a^{3} + 24 a^{2} + 18 a\right)\cdot 31 + \left(29 a^{8} + 5 a^{7} + 25 a^{6} + 17 a^{5} + 5 a^{4} + 28 a^{3} + 12 a^{2} + 15 a + 22\right)\cdot 31^{2} + \left(5 a^{8} + 12 a^{7} + 12 a^{6} + 3 a^{5} + 5 a^{4} + 17 a^{2} + 8 a + 4\right)\cdot 31^{3} + \left(18 a^{8} + 7 a^{6} + 8 a^{5} + 9 a^{4} + 18 a^{3} + 27 a^{2} + 28 a + 9\right)\cdot 31^{4} + \left(27 a^{8} + 24 a^{7} + 29 a^{6} + 9 a^{5} + 20 a^{4} + 6 a^{3} + 11 a + 5\right)\cdot 31^{5} + \left(10 a^{8} + 4 a^{7} + 11 a^{6} + 24 a^{5} + 20 a^{4} + 21 a^{3} + 14 a^{2} + 5 a + 25\right)\cdot 31^{6} + \left(8 a^{8} + 12 a^{7} + 28 a^{6} + 17 a^{5} + 6 a^{4} + 4 a^{3} + 7 a^{2} + 16 a + 28\right)\cdot 31^{7} + \left(a^{8} + 3 a^{7} + 5 a^{6} + 16 a^{5} + 21 a^{4} + 19 a^{3} + 5 a^{2} + 21 a + 28\right)\cdot 31^{8} + \left(25 a^{8} + 29 a^{6} + 3 a^{5} + 19 a^{4} + 7 a^{3} + 21 a^{2} + 26 a + 18\right)\cdot 31^{9} +O(31^{10})$$ 20*a^8 + 9*a^7 + 21*a^6 + 23*a^5 + 28*a^4 + 12*a^2 + 29*a + 2 + (11*a^8 + 22*a^6 + 19*a^5 + 21*a^4 + 13*a^3 + 24*a^2 + 18*a)*31 + (29*a^8 + 5*a^7 + 25*a^6 + 17*a^5 + 5*a^4 + 28*a^3 + 12*a^2 + 15*a + 22)*31^2 + (5*a^8 + 12*a^7 + 12*a^6 + 3*a^5 + 5*a^4 + 17*a^2 + 8*a + 4)*31^3 + (18*a^8 + 7*a^6 + 8*a^5 + 9*a^4 + 18*a^3 + 27*a^2 + 28*a + 9)*31^4 + (27*a^8 + 24*a^7 + 29*a^6 + 9*a^5 + 20*a^4 + 6*a^3 + 11*a + 5)*31^5 + (10*a^8 + 4*a^7 + 11*a^6 + 24*a^5 + 20*a^4 + 21*a^3 + 14*a^2 + 5*a + 25)*31^6 + (8*a^8 + 12*a^7 + 28*a^6 + 17*a^5 + 6*a^4 + 4*a^3 + 7*a^2 + 16*a + 28)*31^7 + (a^8 + 3*a^7 + 5*a^6 + 16*a^5 + 21*a^4 + 19*a^3 + 5*a^2 + 21*a + 28)*31^8 + (25*a^8 + 29*a^6 + 3*a^5 + 19*a^4 + 7*a^3 + 21*a^2 + 26*a + 18)*31^9+O(31^10) $r_{ 11 }$ $=$ $$11 a^{8} + 9 a^{7} + 22 a^{6} + 26 a^{5} + 21 a^{4} + 8 a^{3} + 22 a^{2} + 16 a + \left(25 a^{8} + 20 a^{7} + 30 a^{6} + 10 a^{5} + 11 a^{4} + 30 a^{3} + 6 a^{2} + 27 a + 4\right)\cdot 31 + \left(22 a^{8} + 10 a^{7} + 24 a^{6} + 16 a^{5} + 29 a^{4} + 11 a^{3} + 16 a^{2} + 27 a + 11\right)\cdot 31^{2} + \left(19 a^{8} + 10 a^{7} + 4 a^{6} + 15 a^{5} + 14 a^{4} + 30 a^{3} + 2 a + 7\right)\cdot 31^{3} + \left(a^{8} + 27 a^{7} + 19 a^{6} + 29 a^{5} + 20 a^{3} + 10 a^{2} + 23 a + 5\right)\cdot 31^{4} + \left(9 a^{8} + 13 a^{7} + 28 a^{6} + 30 a^{5} + 6 a^{4} + 7 a^{3} + 25 a + 26\right)\cdot 31^{5} + \left(20 a^{8} + 22 a^{7} + 4 a^{6} + 19 a^{5} + 6 a^{4} + 8 a^{3} + 29 a^{2} + 22 a + 20\right)\cdot 31^{6} + \left(5 a^{8} + 6 a^{7} + 11 a^{6} + 23 a^{5} + 4 a^{4} + 29 a^{3} + 17 a^{2} + a + 28\right)\cdot 31^{7} + \left(8 a^{8} + 22 a^{7} + 19 a^{6} + 18 a^{5} + 29 a^{4} + 15 a^{3} + 18 a^{2} + 16 a + 1\right)\cdot 31^{8} + \left(23 a^{8} + 26 a^{7} + 8 a^{6} + 7 a^{5} + 15 a^{4} + 19 a^{3} + a^{2} + 27 a + 29\right)\cdot 31^{9} +O(31^{10})$$ 11*a^8 + 9*a^7 + 22*a^6 + 26*a^5 + 21*a^4 + 8*a^3 + 22*a^2 + 16*a + (25*a^8 + 20*a^7 + 30*a^6 + 10*a^5 + 11*a^4 + 30*a^3 + 6*a^2 + 27*a + 4)*31 + (22*a^8 + 10*a^7 + 24*a^6 + 16*a^5 + 29*a^4 + 11*a^3 + 16*a^2 + 27*a + 11)*31^2 + (19*a^8 + 10*a^7 + 4*a^6 + 15*a^5 + 14*a^4 + 30*a^3 + 2*a + 7)*31^3 + (a^8 + 27*a^7 + 19*a^6 + 29*a^5 + 20*a^3 + 10*a^2 + 23*a + 5)*31^4 + (9*a^8 + 13*a^7 + 28*a^6 + 30*a^5 + 6*a^4 + 7*a^3 + 25*a + 26)*31^5 + (20*a^8 + 22*a^7 + 4*a^6 + 19*a^5 + 6*a^4 + 8*a^3 + 29*a^2 + 22*a + 20)*31^6 + (5*a^8 + 6*a^7 + 11*a^6 + 23*a^5 + 4*a^4 + 29*a^3 + 17*a^2 + a + 28)*31^7 + (8*a^8 + 22*a^7 + 19*a^6 + 18*a^5 + 29*a^4 + 15*a^3 + 18*a^2 + 16*a + 1)*31^8 + (23*a^8 + 26*a^7 + 8*a^6 + 7*a^5 + 15*a^4 + 19*a^3 + a^2 + 27*a + 29)*31^9+O(31^10) $r_{ 12 }$ $=$ $$18 a^{8} + a^{7} + 30 a^{6} + 16 a^{5} + a^{4} + 30 a^{3} + 20 a^{2} + 26 a + 5 + \left(7 a^{8} + 21 a^{7} + 17 a^{6} + 13 a^{5} + 23 a^{4} + 3 a^{3} + 16 a^{2} + 2 a + 2\right)\cdot 31 + \left(9 a^{8} + 7 a^{7} + 4 a^{6} + 20 a^{5} + 17 a^{4} + 25 a^{3} + 27 a^{2} + 15 a + 22\right)\cdot 31^{2} + \left(9 a^{8} + 5 a^{7} + 30 a^{6} + 14 a^{5} + 23 a^{4} + 30 a^{3} + 14 a^{2} + 4 a + 29\right)\cdot 31^{3} + \left(20 a^{8} + 17 a^{7} + 7 a^{6} + 17 a^{4} + 26 a^{3} + 16 a^{2} + 28 a + 15\right)\cdot 31^{4} + \left(18 a^{8} + 29 a^{7} + 21 a^{6} + a^{5} + 3 a^{4} + 7 a^{3} + 11 a^{2} + 23 a + 22\right)\cdot 31^{5} + \left(9 a^{7} + 14 a^{6} + 17 a^{5} + 13 a^{4} + 24 a^{3} + 11 a^{2} + a + 9\right)\cdot 31^{6} + \left(29 a^{8} + a^{7} + 26 a^{6} + 7 a^{5} + 15 a^{4} + 3 a^{3} + 9 a^{2} + 18 a + 26\right)\cdot 31^{7} + \left(28 a^{8} + 2 a^{7} + 4 a^{6} + 2 a^{5} + 3 a^{4} + 7 a^{3} + 25 a^{2} + 26 a + 1\right)\cdot 31^{8} + \left(15 a^{8} + 21 a^{7} + 25 a^{6} + 15 a^{5} + 2 a^{4} + 22 a^{3} + 4 a^{2} + 28 a + 27\right)\cdot 31^{9} +O(31^{10})$$ 18*a^8 + a^7 + 30*a^6 + 16*a^5 + a^4 + 30*a^3 + 20*a^2 + 26*a + 5 + (7*a^8 + 21*a^7 + 17*a^6 + 13*a^5 + 23*a^4 + 3*a^3 + 16*a^2 + 2*a + 2)*31 + (9*a^8 + 7*a^7 + 4*a^6 + 20*a^5 + 17*a^4 + 25*a^3 + 27*a^2 + 15*a + 22)*31^2 + (9*a^8 + 5*a^7 + 30*a^6 + 14*a^5 + 23*a^4 + 30*a^3 + 14*a^2 + 4*a + 29)*31^3 + (20*a^8 + 17*a^7 + 7*a^6 + 17*a^4 + 26*a^3 + 16*a^2 + 28*a + 15)*31^4 + (18*a^8 + 29*a^7 + 21*a^6 + a^5 + 3*a^4 + 7*a^3 + 11*a^2 + 23*a + 22)*31^5 + (9*a^7 + 14*a^6 + 17*a^5 + 13*a^4 + 24*a^3 + 11*a^2 + a + 9)*31^6 + (29*a^8 + a^7 + 26*a^6 + 7*a^5 + 15*a^4 + 3*a^3 + 9*a^2 + 18*a + 26)*31^7 + (28*a^8 + 2*a^7 + 4*a^6 + 2*a^5 + 3*a^4 + 7*a^3 + 25*a^2 + 26*a + 1)*31^8 + (15*a^8 + 21*a^7 + 25*a^6 + 15*a^5 + 2*a^4 + 22*a^3 + 4*a^2 + 28*a + 27)*31^9+O(31^10) $r_{ 13 }$ $=$ $$26 a^{8} + 15 a^{7} + 27 a^{6} + 2 a^{5} + 14 a^{4} + 9 a^{3} + 8 a^{2} + 10 a + 18 + \left(21 a^{8} + 10 a^{7} + 11 a^{6} + 14 a^{5} + 4 a^{4} + 14 a^{3} + 12 a^{2} + 14 a + 20\right)\cdot 31 + \left(15 a^{8} + 17 a^{7} + 10 a^{6} + 21 a^{5} + 18 a^{4} + 9 a^{3} + 8 a^{2} + 21 a + 23\right)\cdot 31^{2} + \left(a^{8} + 12 a^{7} + 21 a^{6} + 6 a^{5} + 29 a^{4} + 7 a^{3} + 16 a^{2} + 30 a + 1\right)\cdot 31^{3} + \left(22 a^{8} + 13 a^{6} + 11 a^{5} + a^{4} + 17 a^{3} + 15 a^{2} + 15 a + 29\right)\cdot 31^{4} + \left(5 a^{8} + 21 a^{7} + 10 a^{6} + 30 a^{4} + 19 a^{3} + 9 a^{2} + 19 a + 5\right)\cdot 31^{5} + \left(a^{8} + 23 a^{7} + 12 a^{6} + 5 a^{5} + 6 a^{4} + 19 a^{3} + 17 a^{2} + 21 a + 1\right)\cdot 31^{6} + \left(13 a^{8} + 13 a^{7} + 23 a^{6} + 29 a^{5} + 30 a^{4} + 13 a^{3} + 21 a^{2} + 18 a + 27\right)\cdot 31^{7} + \left(23 a^{8} + 10 a^{7} + 3 a^{5} + 20 a^{4} + 6 a^{3} + 19 a^{2} + 23 a + 13\right)\cdot 31^{8} + \left(12 a^{8} + 15 a^{7} + 6 a^{6} + 12 a^{5} + 16 a^{4} + 10 a^{3} + 5 a^{2} + 12 a + 25\right)\cdot 31^{9} +O(31^{10})$$ 26*a^8 + 15*a^7 + 27*a^6 + 2*a^5 + 14*a^4 + 9*a^3 + 8*a^2 + 10*a + 18 + (21*a^8 + 10*a^7 + 11*a^6 + 14*a^5 + 4*a^4 + 14*a^3 + 12*a^2 + 14*a + 20)*31 + (15*a^8 + 17*a^7 + 10*a^6 + 21*a^5 + 18*a^4 + 9*a^3 + 8*a^2 + 21*a + 23)*31^2 + (a^8 + 12*a^7 + 21*a^6 + 6*a^5 + 29*a^4 + 7*a^3 + 16*a^2 + 30*a + 1)*31^3 + (22*a^8 + 13*a^6 + 11*a^5 + a^4 + 17*a^3 + 15*a^2 + 15*a + 29)*31^4 + (5*a^8 + 21*a^7 + 10*a^6 + 30*a^4 + 19*a^3 + 9*a^2 + 19*a + 5)*31^5 + (a^8 + 23*a^7 + 12*a^6 + 5*a^5 + 6*a^4 + 19*a^3 + 17*a^2 + 21*a + 1)*31^6 + (13*a^8 + 13*a^7 + 23*a^6 + 29*a^5 + 30*a^4 + 13*a^3 + 21*a^2 + 18*a + 27)*31^7 + (23*a^8 + 10*a^7 + 3*a^5 + 20*a^4 + 6*a^3 + 19*a^2 + 23*a + 13)*31^8 + (12*a^8 + 15*a^7 + 6*a^6 + 12*a^5 + 16*a^4 + 10*a^3 + 5*a^2 + 12*a + 25)*31^9+O(31^10) $r_{ 14 }$ $=$ $$14 a^{8} + 21 a^{7} + 26 a^{6} + 16 a^{5} + 20 a^{4} + 21 a^{3} + 21 a^{2} + 23 a + 6 + \left(27 a^{8} + 20 a^{7} + 11 a^{6} + 9 a^{5} + 27 a^{4} + 9 a^{3} + 7 a^{2} + 9 a + 17\right)\cdot 31 + \left(27 a^{8} + 12 a^{7} + 9 a^{6} + 4 a^{5} + 18 a^{4} + 12 a^{3} + 29 a^{2} + 15 a\right)\cdot 31^{2} + \left(7 a^{8} + 26 a^{7} + 5 a^{6} + 4 a^{5} + a^{4} + 27 a^{3} + 17 a^{2} + 29 a + 12\right)\cdot 31^{3} + \left(27 a^{8} + 30 a^{7} + 9 a^{6} + a^{5} + 7 a^{4} + 13 a^{3} + 10 a^{2} + 9 a + 1\right)\cdot 31^{4} + \left(15 a^{8} + 26 a^{7} + 24 a^{6} + a^{5} + 18 a^{4} + 25 a^{3} + 16 a^{2} + 15 a + 26\right)\cdot 31^{5} + \left(3 a^{8} + 6 a^{7} + 18 a^{6} + 17 a^{5} + 25 a^{4} + 6 a^{3} + 13 a^{2} + 12 a + 9\right)\cdot 31^{6} + \left(8 a^{8} + 16 a^{7} + 7 a^{6} + 20 a^{5} + 17 a^{3} + 9 a^{2} + 18 a + 13\right)\cdot 31^{7} + \left(4 a^{8} + a^{7} + 20 a^{6} + 12 a^{5} + 27 a^{4} + 14 a^{2} + 14 a + 8\right)\cdot 31^{8} + \left(19 a^{7} + 15 a^{6} + 23 a^{5} + a^{4} + 20 a^{3} + 7 a^{2} + 15 a + 21\right)\cdot 31^{9} +O(31^{10})$$ 14*a^8 + 21*a^7 + 26*a^6 + 16*a^5 + 20*a^4 + 21*a^3 + 21*a^2 + 23*a + 6 + (27*a^8 + 20*a^7 + 11*a^6 + 9*a^5 + 27*a^4 + 9*a^3 + 7*a^2 + 9*a + 17)*31 + (27*a^8 + 12*a^7 + 9*a^6 + 4*a^5 + 18*a^4 + 12*a^3 + 29*a^2 + 15*a)*31^2 + (7*a^8 + 26*a^7 + 5*a^6 + 4*a^5 + a^4 + 27*a^3 + 17*a^2 + 29*a + 12)*31^3 + (27*a^8 + 30*a^7 + 9*a^6 + a^5 + 7*a^4 + 13*a^3 + 10*a^2 + 9*a + 1)*31^4 + (15*a^8 + 26*a^7 + 24*a^6 + a^5 + 18*a^4 + 25*a^3 + 16*a^2 + 15*a + 26)*31^5 + (3*a^8 + 6*a^7 + 18*a^6 + 17*a^5 + 25*a^4 + 6*a^3 + 13*a^2 + 12*a + 9)*31^6 + (8*a^8 + 16*a^7 + 7*a^6 + 20*a^5 + 17*a^3 + 9*a^2 + 18*a + 13)*31^7 + (4*a^8 + a^7 + 20*a^6 + 12*a^5 + 27*a^4 + 14*a^2 + 14*a + 8)*31^8 + (19*a^7 + 15*a^6 + 23*a^5 + a^4 + 20*a^3 + 7*a^2 + 15*a + 21)*31^9+O(31^10) $r_{ 15 }$ $=$ $$20 a^{8} + 26 a^{7} + 26 a^{6} + 27 a^{5} + 14 a^{4} + 24 a^{3} + 8 a^{2} + 21 a + 18 + \left(19 a^{8} + 18 a^{7} + 7 a^{6} + 6 a^{5} + 23 a^{4} + 26 a^{3} + 21 a^{2} + 2 a\right)\cdot 31 + \left(8 a^{8} + 5 a^{7} + 29 a^{6} + 18 a^{5} + 19 a^{4} + 13 a^{3} + 4 a^{2} + 26 a + 2\right)\cdot 31^{2} + \left(14 a^{8} + 13 a^{7} + 15 a^{6} + 18 a^{4} + 8 a^{3} + 29 a^{2} + 9 a + 30\right)\cdot 31^{3} + \left(5 a^{8} + 2 a^{7} + 19 a^{6} + 12 a^{5} + 23 a^{4} + 18 a^{3} + 27 a^{2} + 13 a + 16\right)\cdot 31^{4} + \left(5 a^{8} + 8 a^{7} + 8 a^{6} + 24 a^{4} + 22 a^{3} + 30 a^{2} + 18 a + 6\right)\cdot 31^{5} + \left(13 a^{8} + 29 a^{7} + 9 a^{6} + 2 a^{5} + 11 a^{4} + 21 a^{3} + 24 a^{2} + 20 a + 20\right)\cdot 31^{6} + \left(4 a^{8} + 19 a^{7} + 26 a^{6} + 23 a^{5} + 24 a^{4} + 22 a^{3} + 7 a^{2} + 11 a + 15\right)\cdot 31^{7} + \left(7 a^{8} + 30 a^{7} + 22 a^{5} + 11 a^{4} + 18 a^{3} + 24 a^{2} + 24 a + 15\right)\cdot 31^{8} + \left(3 a^{8} + 2 a^{7} + 15 a^{6} + 12 a^{5} + 24 a^{4} + 11 a^{3} + 26 a^{2} + 21 a + 4\right)\cdot 31^{9} +O(31^{10})$$ 20*a^8 + 26*a^7 + 26*a^6 + 27*a^5 + 14*a^4 + 24*a^3 + 8*a^2 + 21*a + 18 + (19*a^8 + 18*a^7 + 7*a^6 + 6*a^5 + 23*a^4 + 26*a^3 + 21*a^2 + 2*a)*31 + (8*a^8 + 5*a^7 + 29*a^6 + 18*a^5 + 19*a^4 + 13*a^3 + 4*a^2 + 26*a + 2)*31^2 + (14*a^8 + 13*a^7 + 15*a^6 + 18*a^4 + 8*a^3 + 29*a^2 + 9*a + 30)*31^3 + (5*a^8 + 2*a^7 + 19*a^6 + 12*a^5 + 23*a^4 + 18*a^3 + 27*a^2 + 13*a + 16)*31^4 + (5*a^8 + 8*a^7 + 8*a^6 + 24*a^4 + 22*a^3 + 30*a^2 + 18*a + 6)*31^5 + (13*a^8 + 29*a^7 + 9*a^6 + 2*a^5 + 11*a^4 + 21*a^3 + 24*a^2 + 20*a + 20)*31^6 + (4*a^8 + 19*a^7 + 26*a^6 + 23*a^5 + 24*a^4 + 22*a^3 + 7*a^2 + 11*a + 15)*31^7 + (7*a^8 + 30*a^7 + 22*a^5 + 11*a^4 + 18*a^3 + 24*a^2 + 24*a + 15)*31^8 + (3*a^8 + 2*a^7 + 15*a^6 + 12*a^5 + 24*a^4 + 11*a^3 + 26*a^2 + 21*a + 4)*31^9+O(31^10) $r_{ 16 }$ $=$ $$4 a^{8} + 8 a^{7} + 29 a^{6} + 17 a^{5} + 18 a^{4} + 21 a^{2} + 12 a + 19 + \left(22 a^{8} + 27 a^{7} + 3 a^{6} + 24 a^{5} + 3 a^{4} + 15 a^{3} + 26 a^{2} + 7 a + 2\right)\cdot 31 + \left(17 a^{8} + 28 a^{7} + 16 a^{6} + 26 a^{5} + 10 a^{4} + 19 a^{3} + 6 a^{2} + 30 a + 5\right)\cdot 31^{2} + \left(12 a^{8} + 25 a^{7} + 28 a^{6} + 3 a^{5} + 7 a^{4} + 28 a^{3} + 27 a^{2} + 4 a + 18\right)\cdot 31^{3} + \left(13 a^{8} + 7 a^{7} + 24 a^{6} + 22 a^{5} + 17 a^{4} + 22 a^{3} + 19 a^{2} + 27 a + 27\right)\cdot 31^{4} + \left(24 a^{8} + 27 a^{7} + 18 a^{6} + 7 a^{5} + 16 a^{4} + 30 a^{3} + 6 a^{2} + 5 a + 18\right)\cdot 31^{5} + \left(13 a^{8} + 23 a^{7} + 30 a^{6} + 5 a^{5} + 6 a^{4} + a^{3} + 11 a^{2} + 14 a + 26\right)\cdot 31^{6} + \left(18 a^{8} + 9 a^{7} + 4 a^{6} + 20 a^{5} + a^{4} + 10 a^{3} + 14 a^{2} + 20 a + 13\right)\cdot 31^{7} + \left(12 a^{8} + 8 a^{7} + 20 a^{6} + 25 a^{5} + 28 a^{4} + 29 a^{3} + 5 a^{2} + 17 a + 1\right)\cdot 31^{8} + \left(9 a^{8} + 25 a^{7} + 15 a^{6} + 23 a^{5} + 15 a^{3} + 7 a^{2} + 24 a + 9\right)\cdot 31^{9} +O(31^{10})$$ 4*a^8 + 8*a^7 + 29*a^6 + 17*a^5 + 18*a^4 + 21*a^2 + 12*a + 19 + (22*a^8 + 27*a^7 + 3*a^6 + 24*a^5 + 3*a^4 + 15*a^3 + 26*a^2 + 7*a + 2)*31 + (17*a^8 + 28*a^7 + 16*a^6 + 26*a^5 + 10*a^4 + 19*a^3 + 6*a^2 + 30*a + 5)*31^2 + (12*a^8 + 25*a^7 + 28*a^6 + 3*a^5 + 7*a^4 + 28*a^3 + 27*a^2 + 4*a + 18)*31^3 + (13*a^8 + 7*a^7 + 24*a^6 + 22*a^5 + 17*a^4 + 22*a^3 + 19*a^2 + 27*a + 27)*31^4 + (24*a^8 + 27*a^7 + 18*a^6 + 7*a^5 + 16*a^4 + 30*a^3 + 6*a^2 + 5*a + 18)*31^5 + (13*a^8 + 23*a^7 + 30*a^6 + 5*a^5 + 6*a^4 + a^3 + 11*a^2 + 14*a + 26)*31^6 + (18*a^8 + 9*a^7 + 4*a^6 + 20*a^5 + a^4 + 10*a^3 + 14*a^2 + 20*a + 13)*31^7 + (12*a^8 + 8*a^7 + 20*a^6 + 25*a^5 + 28*a^4 + 29*a^3 + 5*a^2 + 17*a + 1)*31^8 + (9*a^8 + 25*a^7 + 15*a^6 + 23*a^5 + 15*a^3 + 7*a^2 + 24*a + 9)*31^9+O(31^10) $r_{ 17 }$ $=$ $$27 a^{8} + 8 a^{7} + 23 a^{6} + 30 a^{5} + 29 a^{4} + 14 a^{3} + 23 a^{2} + 28 a + \left(3 a^{8} + 3 a^{6} + 4 a^{5} + 7 a^{4} + 16 a^{3} + 23 a^{2} + 13 a + 7\right)\cdot 31 + \left(19 a^{8} + 25 a^{7} + 8 a^{6} + 25 a^{5} + 24 a^{4} + 25 a^{3} + 24 a^{2} + 24 a + 1\right)\cdot 31^{2} + \left(12 a^{8} + 4 a^{7} + 10 a^{6} + 15 a^{5} + 11 a^{4} + 6 a^{3} + 21 a^{2} + 12 a + 14\right)\cdot 31^{3} + \left(5 a^{8} + 10 a^{7} + 2 a^{6} + 22 a^{5} + 6 a^{4} + 15 a^{3} + 23 a^{2} + 7 a\right)\cdot 31^{4} + \left(10 a^{8} + 7 a^{6} + 25 a^{5} + a^{4} + 11 a^{3} + 10 a^{2} + 6 a + 21\right)\cdot 31^{5} + \left(2 a^{8} + 17 a^{7} + 20 a^{6} + 18 a^{4} + 19 a^{3} + 15 a^{2} + 14 a + 30\right)\cdot 31^{6} + \left(5 a^{8} + 19 a^{7} + 2 a^{6} + 4 a^{4} + 29 a^{3} + 23 a^{2} + 16 a + 7\right)\cdot 31^{7} + \left(16 a^{8} + 12 a^{7} + 11 a^{6} + 18 a^{5} + 20 a^{4} + 7 a^{3} + 10 a^{2} + 2 a + 19\right)\cdot 31^{8} + \left(a^{8} + 2 a^{7} + 13 a^{6} + 8 a^{5} + 21 a^{4} + 20 a^{3} + 23 a^{2} + 3 a + 29\right)\cdot 31^{9} +O(31^{10})$$ 27*a^8 + 8*a^7 + 23*a^6 + 30*a^5 + 29*a^4 + 14*a^3 + 23*a^2 + 28*a + (3*a^8 + 3*a^6 + 4*a^5 + 7*a^4 + 16*a^3 + 23*a^2 + 13*a + 7)*31 + (19*a^8 + 25*a^7 + 8*a^6 + 25*a^5 + 24*a^4 + 25*a^3 + 24*a^2 + 24*a + 1)*31^2 + (12*a^8 + 4*a^7 + 10*a^6 + 15*a^5 + 11*a^4 + 6*a^3 + 21*a^2 + 12*a + 14)*31^3 + (5*a^8 + 10*a^7 + 2*a^6 + 22*a^5 + 6*a^4 + 15*a^3 + 23*a^2 + 7*a)*31^4 + (10*a^8 + 7*a^6 + 25*a^5 + a^4 + 11*a^3 + 10*a^2 + 6*a + 21)*31^5 + (2*a^8 + 17*a^7 + 20*a^6 + 18*a^4 + 19*a^3 + 15*a^2 + 14*a + 30)*31^6 + (5*a^8 + 19*a^7 + 2*a^6 + 4*a^4 + 29*a^3 + 23*a^2 + 16*a + 7)*31^7 + (16*a^8 + 12*a^7 + 11*a^6 + 18*a^5 + 20*a^4 + 7*a^3 + 10*a^2 + 2*a + 19)*31^8 + (a^8 + 2*a^7 + 13*a^6 + 8*a^5 + 21*a^4 + 20*a^3 + 23*a^2 + 3*a + 29)*31^9+O(31^10) $r_{ 18 }$ $=$ $$16 a^{8} + 25 a^{7} + 6 a^{6} + 28 a^{5} + 25 a^{4} + 6 a^{3} + 4 a^{2} + 30 a + 1 + \left(12 a^{8} + 14 a^{7} + 30 a^{6} + 4 a^{5} + 2 a^{4} + 21 a^{3} + 23 a^{2} + 21 a + 10\right)\cdot 31 + \left(26 a^{8} + 7 a^{7} + 23 a^{6} + 17 a^{5} + 12 a^{4} + 3 a^{3} + 20 a^{2} + 20 a + 27\right)\cdot 31^{2} + \left(15 a^{8} + 30 a^{7} + 13 a^{6} + 10 a^{5} + 10 a^{4} + 27 a^{3} + 19 a^{2} + 12 a + 9\right)\cdot 31^{3} + \left(9 a^{8} + 10 a^{7} + 7 a^{6} + 21 a^{5} + 20 a^{4} + 29 a^{3} + 26 a^{2} + 29\right)\cdot 31^{4} + \left(12 a^{8} + 10 a^{7} + 8 a^{6} + 4 a^{5} + 4 a^{4} + 17 a^{3} + 9 a^{2} + 10 a + 24\right)\cdot 31^{5} + \left(6 a^{8} + 2 a^{7} + 20 a^{5} + 9 a^{4} + 14 a^{3} + 26 a + 22\right)\cdot 31^{6} + \left(25 a^{8} + 21 a^{6} + 10 a^{5} + 24 a^{3} + 7 a^{2} + 20 a + 7\right)\cdot 31^{7} + \left(9 a^{8} + 21 a^{7} + 23 a^{6} + a^{5} + 8 a^{4} + 14 a^{3} + 11 a^{2} + 25 a + 26\right)\cdot 31^{8} + \left(12 a^{8} + 2 a^{7} + a^{6} + 30 a^{5} + 14 a^{4} + 6 a^{3} + 26 a^{2} + 13 a + 22\right)\cdot 31^{9} +O(31^{10})$$ 16*a^8 + 25*a^7 + 6*a^6 + 28*a^5 + 25*a^4 + 6*a^3 + 4*a^2 + 30*a + 1 + (12*a^8 + 14*a^7 + 30*a^6 + 4*a^5 + 2*a^4 + 21*a^3 + 23*a^2 + 21*a + 10)*31 + (26*a^8 + 7*a^7 + 23*a^6 + 17*a^5 + 12*a^4 + 3*a^3 + 20*a^2 + 20*a + 27)*31^2 + (15*a^8 + 30*a^7 + 13*a^6 + 10*a^5 + 10*a^4 + 27*a^3 + 19*a^2 + 12*a + 9)*31^3 + (9*a^8 + 10*a^7 + 7*a^6 + 21*a^5 + 20*a^4 + 29*a^3 + 26*a^2 + 29)*31^4 + (12*a^8 + 10*a^7 + 8*a^6 + 4*a^5 + 4*a^4 + 17*a^3 + 9*a^2 + 10*a + 24)*31^5 + (6*a^8 + 2*a^7 + 20*a^5 + 9*a^4 + 14*a^3 + 26*a + 22)*31^6 + (25*a^8 + 21*a^6 + 10*a^5 + 24*a^3 + 7*a^2 + 20*a + 7)*31^7 + (9*a^8 + 21*a^7 + 23*a^6 + a^5 + 8*a^4 + 14*a^3 + 11*a^2 + 25*a + 26)*31^8 + (12*a^8 + 2*a^7 + a^6 + 30*a^5 + 14*a^4 + 6*a^3 + 26*a^2 + 13*a + 22)*31^9+O(31^10)

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

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 18 }$ Character value $1$ $1$ $()$ $2$ $3$ $2$ $(1,8)(2,7)(3,4)(5,6)(9,16)(10,15)(11,18)(12,17)(13,14)$ $0$ $1$ $3$ $(1,7,13)(2,14,8)(3,9,15)(4,16,10)(5,17,11)(6,12,18)$ $2 \zeta_{9}^{3}$ $1$ $3$ $(1,13,7)(2,8,14)(3,15,9)(4,10,16)(5,11,17)(6,18,12)$ $-2 \zeta_{9}^{3} - 2$ $2$ $3$ $(1,7,13)(2,8,14)(3,9,15)(4,10,16)(5,11,17)(6,12,18)$ $-1$ $2$ $3$ $(1,7,13)(3,9,15)(6,12,18)$ $\zeta_{9}^{3} + 1$ $2$ $3$ $(1,13,7)(3,15,9)(6,18,12)$ $-\zeta_{9}^{3}$ $3$ $6$ $(1,2,13,8,7,14)(3,16,15,4,9,10)(5,12,11,6,17,18)$ $0$ $3$ $6$ $(1,14,7,8,13,2)(3,10,9,4,15,16)(5,18,17,6,11,12)$ $0$ $1$ $9$ $(1,15,6,7,3,12,13,9,18)(2,4,17,14,16,11,8,10,5)$ $2 \zeta_{9}^{4}$ $1$ $9$ $(1,6,3,13,18,15,7,12,9)(2,17,16,8,5,4,14,11,10)$ $-2 \zeta_{9}^{5} - 2 \zeta_{9}^{2}$ $1$ $9$ $(1,3,18,7,9,6,13,15,12)(2,16,5,14,10,17,8,4,11)$ $-2 \zeta_{9}^{4} - 2 \zeta_{9}$ $1$ $9$ $(1,12,15,13,6,9,7,18,3)(2,11,4,8,17,10,14,5,16)$ $2 \zeta_{9}^{2}$ $1$ $9$ $(1,9,12,7,15,18,13,3,6)(2,10,11,14,4,5,8,16,17)$ $2 \zeta_{9}$ $1$ $9$ $(1,18,9,13,12,3,7,6,15)(2,5,10,8,11,16,14,17,4)$ $2 \zeta_{9}^{5}$ $2$ $9$ $(1,3,18,7,9,6,13,15,12)(2,10,11,14,4,5,8,16,17)$ $-\zeta_{9}^{4}$ $2$ $9$ $(1,18,9,13,12,3,7,6,15)(2,11,4,8,17,10,14,5,16)$ $\zeta_{9}^{5} + \zeta_{9}^{2}$ $2$ $9$ $(1,9,12,7,15,18,13,3,6)(2,4,17,14,16,11,8,10,5)$ $\zeta_{9}^{4} + \zeta_{9}$ $2$ $9$ $(1,6,3,13,18,15,7,12,9)(2,5,10,8,11,16,14,17,4)$ $-\zeta_{9}^{2}$ $2$ $9$ $(1,15,6,7,3,12,13,9,18)(2,16,5,14,10,17,8,4,11)$ $-\zeta_{9}$ $2$ $9$ $(1,12,15,13,6,9,7,18,3)(2,17,16,8,5,4,14,11,10)$ $-\zeta_{9}^{5}$ $3$ $18$ $(1,10,6,2,3,17,13,16,18,8,15,5,7,4,12,14,9,11)$ $0$ $3$ $18$ $(1,17,15,14,6,16,7,11,3,8,12,10,13,5,9,2,18,4)$ $0$ $3$ $18$ $(1,16,12,2,15,11,13,4,6,8,9,17,7,10,18,14,3,5)$ $0$ $3$ $18$ $(1,5,3,14,18,10,7,17,9,8,6,4,13,11,15,2,12,16)$ $0$ $3$ $18$ $(1,4,18,2,9,5,13,10,12,8,3,11,7,16,6,14,15,17)$ $0$ $3$ $18$ $(1,11,9,14,12,4,7,5,15,8,18,16,13,17,3,2,6,10)$ $0$

The blue line marks the conjugacy class containing complex conjugation.