# Properties

 Label 16.604...088.17t5.a.a Dimension $16$ Group $F_{17}$ Conductor $6.045\times 10^{23}$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $16$ Group: $F_{17}$ Conductor: $$604\!\cdots\!088$$$$\medspace = 2^{79}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 17.1.604462909807314587353088.1 Galois orbit size: $1$ Smallest permutation container: $F_{17}$ Parity: even Determinant: 1.8.2t1.a.a Projective image: $F_{17}$ Projective stem field: Galois closure of 17.1.604462909807314587353088.1

## Defining polynomial

 $f(x)$ $=$ $$x^{17} - 2 x^{16} + 8 x^{13} + 16 x^{12} - 16 x^{11} + 64 x^{9} - 32 x^{8} - 80 x^{7} + 32 x^{6} + 40 x^{5} + 80 x^{4} + 16 x^{3} - 128 x^{2} - 2 x + 68$$ x^17 - 2*x^16 + 8*x^13 + 16*x^12 - 16*x^11 + 64*x^9 - 32*x^8 - 80*x^7 + 32*x^6 + 40*x^5 + 80*x^4 + 16*x^3 - 128*x^2 - 2*x + 68 .

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $$x^{8} + 3x^{4} + 20x^{3} + 5x^{2} + 3x + 5$$

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

## Generators of the action on the roots $r_1, \ldots, r_{ 17 }$

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.