# Properties

 Label 2.171.16t60.a.a Dimension $2$ Group $\SL(2,3):C_2$ Conductor $171$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: 16T60 Conductor: $$171$$$$\medspace = 3^{2} \cdot 19$$ Artin stem field: Galois closure of 16.0.731086699811838561.1 Galois orbit size: $4$ Smallest permutation container: 16T60 Parity: odd Determinant: 1.171.6t1.f.b Projective image: $A_4$ Projective stem field: Galois closure of 4.0.29241.1

## Defining polynomial

 $f(x)$ $=$ $$x^{16} - 3 x^{15} + 6 x^{14} - 4 x^{13} - x^{12} + 15 x^{11} - 27 x^{10} + 32 x^{9} - 12 x^{8} - 23 x^{7} + 55 x^{6} - 57 x^{5} + 43 x^{4} - 23 x^{3} + 9 x^{2} - x + 1$$ x^16 - 3*x^15 + 6*x^14 - 4*x^13 - x^12 + 15*x^11 - 27*x^10 + 32*x^9 - 12*x^8 - 23*x^7 + 55*x^6 - 57*x^5 + 43*x^4 - 23*x^3 + 9*x^2 - x + 1 .

The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 9.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $$x^{6} + 19x^{3} + 28x^{2} + 21x + 3$$

Roots:
 $r_{ 1 }$ $=$ $$24 a^{5} + 11 a^{4} + a^{3} + 15 a^{2} + 14 a + 18 + \left(23 a^{5} + 20 a^{4} + 22 a^{3} + 30 a^{2} + 7 a + 18\right)\cdot 43 + \left(25 a^{5} + 34 a^{4} + 3 a^{3} + 42 a^{2} + 20 a + 42\right)\cdot 43^{2} + \left(42 a^{5} + 16 a^{4} + 14 a^{3} + 18 a^{2} + 38 a + 7\right)\cdot 43^{3} + \left(4 a^{5} + 9 a^{4} + 31 a^{3} + 24 a^{2} + 26 a + 15\right)\cdot 43^{4} + \left(33 a^{5} + 39 a^{4} + 3 a^{3} + 2 a^{2} + 13 a + 35\right)\cdot 43^{5} + \left(33 a^{5} + 23 a^{4} + 25 a^{3} + 35 a^{2} + 32 a + 36\right)\cdot 43^{6} + \left(9 a^{5} + 19 a^{4} + 12 a^{3} + 8 a^{2} + 32 a + 1\right)\cdot 43^{7} + \left(2 a^{5} + 23 a^{4} + 38 a^{3} + 13 a^{2} + 11 a + 13\right)\cdot 43^{8} +O(43^{9})$$ 24*a^5 + 11*a^4 + a^3 + 15*a^2 + 14*a + 18 + (23*a^5 + 20*a^4 + 22*a^3 + 30*a^2 + 7*a + 18)*43 + (25*a^5 + 34*a^4 + 3*a^3 + 42*a^2 + 20*a + 42)*43^2 + (42*a^5 + 16*a^4 + 14*a^3 + 18*a^2 + 38*a + 7)*43^3 + (4*a^5 + 9*a^4 + 31*a^3 + 24*a^2 + 26*a + 15)*43^4 + (33*a^5 + 39*a^4 + 3*a^3 + 2*a^2 + 13*a + 35)*43^5 + (33*a^5 + 23*a^4 + 25*a^3 + 35*a^2 + 32*a + 36)*43^6 + (9*a^5 + 19*a^4 + 12*a^3 + 8*a^2 + 32*a + 1)*43^7 + (2*a^5 + 23*a^4 + 38*a^3 + 13*a^2 + 11*a + 13)*43^8+O(43^9) $r_{ 2 }$ $=$ $$16 a^{5} + 33 a^{4} + 5 a^{3} + 35 a^{2} + 18 a + 2 + \left(38 a^{5} + 33 a^{4} + 9 a^{3} + 30 a^{2} + 39 a + 41\right)\cdot 43 + \left(21 a^{5} + 19 a^{4} + 24 a^{3} + 14 a^{2} + 22 a + 10\right)\cdot 43^{2} + \left(37 a^{5} + 4 a^{3} + 33 a^{2} + 5 a + 38\right)\cdot 43^{3} + \left(27 a^{5} + 38 a^{4} + 32 a^{3} + 34 a^{2} + 15 a + 42\right)\cdot 43^{4} + \left(23 a^{5} + 32 a^{4} + a^{3} + 35 a^{2} + 8 a + 33\right)\cdot 43^{5} + \left(24 a^{5} + 28 a^{4} + 29 a^{3} + 22 a^{2} + 22 a + 10\right)\cdot 43^{6} + \left(38 a^{5} + 34 a^{4} + 30 a^{3} + 31 a^{2} + 10 a + 31\right)\cdot 43^{7} + \left(26 a^{5} + 8 a^{4} + 3 a^{3} + 19 a^{2} + 25 a + 23\right)\cdot 43^{8} +O(43^{9})$$ 16*a^5 + 33*a^4 + 5*a^3 + 35*a^2 + 18*a + 2 + (38*a^5 + 33*a^4 + 9*a^3 + 30*a^2 + 39*a + 41)*43 + (21*a^5 + 19*a^4 + 24*a^3 + 14*a^2 + 22*a + 10)*43^2 + (37*a^5 + 4*a^3 + 33*a^2 + 5*a + 38)*43^3 + (27*a^5 + 38*a^4 + 32*a^3 + 34*a^2 + 15*a + 42)*43^4 + (23*a^5 + 32*a^4 + a^3 + 35*a^2 + 8*a + 33)*43^5 + (24*a^5 + 28*a^4 + 29*a^3 + 22*a^2 + 22*a + 10)*43^6 + (38*a^5 + 34*a^4 + 30*a^3 + 31*a^2 + 10*a + 31)*43^7 + (26*a^5 + 8*a^4 + 3*a^3 + 19*a^2 + 25*a + 23)*43^8+O(43^9) $r_{ 3 }$ $=$ $$5 a^{5} + 4 a^{4} + 39 a^{3} + 41 a^{2} + 38 a + 11 + \left(16 a^{5} + 28 a^{4} + 10 a^{3} + 5 a^{2} + 30 a\right)\cdot 43 + \left(a^{5} + 39 a^{4} + a^{3} + 26 a^{2} + 31 a\right)\cdot 43^{2} + \left(2 a^{5} + 40 a^{4} + 14 a^{3} + 13 a + 5\right)\cdot 43^{3} + \left(27 a^{5} + 11 a^{4} + 13 a^{3} + 21 a^{2} + 41 a + 42\right)\cdot 43^{4} + \left(21 a^{5} + 12 a^{4} + 37 a^{3} + 6 a^{2} + 11 a + 24\right)\cdot 43^{5} + \left(33 a^{5} + 13 a^{4} + 22 a^{3} + 41 a^{2} + 22 a + 23\right)\cdot 43^{6} + \left(27 a^{5} + 39 a^{4} + 24 a^{3} + 13 a^{2} + 21 a + 8\right)\cdot 43^{7} + \left(22 a^{5} + 18 a^{4} + 2 a^{3} + 4 a^{2} + 36 a + 12\right)\cdot 43^{8} +O(43^{9})$$ 5*a^5 + 4*a^4 + 39*a^3 + 41*a^2 + 38*a + 11 + (16*a^5 + 28*a^4 + 10*a^3 + 5*a^2 + 30*a)*43 + (a^5 + 39*a^4 + a^3 + 26*a^2 + 31*a)*43^2 + (2*a^5 + 40*a^4 + 14*a^3 + 13*a + 5)*43^3 + (27*a^5 + 11*a^4 + 13*a^3 + 21*a^2 + 41*a + 42)*43^4 + (21*a^5 + 12*a^4 + 37*a^3 + 6*a^2 + 11*a + 24)*43^5 + (33*a^5 + 13*a^4 + 22*a^3 + 41*a^2 + 22*a + 23)*43^6 + (27*a^5 + 39*a^4 + 24*a^3 + 13*a^2 + 21*a + 8)*43^7 + (22*a^5 + 18*a^4 + 2*a^3 + 4*a^2 + 36*a + 12)*43^8+O(43^9) $r_{ 4 }$ $=$ $$21 a^{5} + 39 a^{4} + 8 a^{3} + 5 a^{2} + 4 a + 10 + \left(41 a^{5} + 39 a^{4} + 41 a^{3} + 14 a^{2} + 9 a + 28\right)\cdot 43 + \left(42 a^{5} + 34 a^{4} + 11 a^{3} + 20 a^{2} + 36 a + 34\right)\cdot 43^{2} + \left(15 a^{5} + 7 a^{4} + 29 a^{3} + 23 a^{2} + 31 a + 20\right)\cdot 43^{3} + \left(21 a^{5} + 22 a^{4} + 29 a^{3} + 42 a^{2} + 33 a + 30\right)\cdot 43^{4} + \left(36 a^{5} + 41 a^{4} + 2 a^{3} + 34 a^{2} + 20 a + 5\right)\cdot 43^{5} + \left(29 a^{5} + 28 a^{4} + 15 a^{3} + 29 a^{2} + 6 a + 3\right)\cdot 43^{6} + \left(14 a^{5} + 26 a^{4} + 37 a^{3} + 25 a^{2} + 36 a + 33\right)\cdot 43^{7} + \left(8 a^{5} + 23 a^{4} + 24 a^{3} + 13 a^{2} + 31 a + 2\right)\cdot 43^{8} +O(43^{9})$$ 21*a^5 + 39*a^4 + 8*a^3 + 5*a^2 + 4*a + 10 + (41*a^5 + 39*a^4 + 41*a^3 + 14*a^2 + 9*a + 28)*43 + (42*a^5 + 34*a^4 + 11*a^3 + 20*a^2 + 36*a + 34)*43^2 + (15*a^5 + 7*a^4 + 29*a^3 + 23*a^2 + 31*a + 20)*43^3 + (21*a^5 + 22*a^4 + 29*a^3 + 42*a^2 + 33*a + 30)*43^4 + (36*a^5 + 41*a^4 + 2*a^3 + 34*a^2 + 20*a + 5)*43^5 + (29*a^5 + 28*a^4 + 15*a^3 + 29*a^2 + 6*a + 3)*43^6 + (14*a^5 + 26*a^4 + 37*a^3 + 25*a^2 + 36*a + 33)*43^7 + (8*a^5 + 23*a^4 + 24*a^3 + 13*a^2 + 31*a + 2)*43^8+O(43^9) $r_{ 5 }$ $=$ $$22 a^{5} + 16 a^{4} + 15 a^{3} + 16 a^{2} + 32 a + 23 + \left(14 a^{5} + 2 a^{4} + 37 a^{3} + 3 a^{2} + 6 a + 1\right)\cdot 43 + \left(35 a^{5} + 3 a^{4} + 8 a^{3} + 21 a^{2} + 21 a + 15\right)\cdot 43^{2} + \left(19 a^{5} + 28 a^{4} + 7 a^{3} + 18 a^{2} + 11 a + 5\right)\cdot 43^{3} + \left(6 a^{5} + 13 a^{4} + 29 a^{3} + 29 a^{2} + 9 a + 21\right)\cdot 43^{4} + \left(13 a^{5} + 21 a^{4} + 13 a^{3} + 22 a^{2} + 42 a + 12\right)\cdot 43^{5} + \left(11 a^{5} + 35 a^{4} + 22 a^{3} + 42 a^{2} + 23 a + 34\right)\cdot 43^{6} + \left(34 a^{5} + 5 a^{4} + 22 a^{3} + 28 a^{2} + 32 a + 31\right)\cdot 43^{7} + \left(14 a^{5} + 14 a^{4} + 19 a^{3} + 19 a^{2} + 21 a + 9\right)\cdot 43^{8} +O(43^{9})$$ 22*a^5 + 16*a^4 + 15*a^3 + 16*a^2 + 32*a + 23 + (14*a^5 + 2*a^4 + 37*a^3 + 3*a^2 + 6*a + 1)*43 + (35*a^5 + 3*a^4 + 8*a^3 + 21*a^2 + 21*a + 15)*43^2 + (19*a^5 + 28*a^4 + 7*a^3 + 18*a^2 + 11*a + 5)*43^3 + (6*a^5 + 13*a^4 + 29*a^3 + 29*a^2 + 9*a + 21)*43^4 + (13*a^5 + 21*a^4 + 13*a^3 + 22*a^2 + 42*a + 12)*43^5 + (11*a^5 + 35*a^4 + 22*a^3 + 42*a^2 + 23*a + 34)*43^6 + (34*a^5 + 5*a^4 + 22*a^3 + 28*a^2 + 32*a + 31)*43^7 + (14*a^5 + 14*a^4 + 19*a^3 + 19*a^2 + 21*a + 9)*43^8+O(43^9) $r_{ 6 }$ $=$ $$19 a^{5} + 20 a^{4} + 28 a^{3} + 32 a^{2} + 24 a + 16 + \left(10 a^{5} + 41 a^{4} + 26 a^{3} + 26 a^{2} + 6 a + 27\right)\cdot 43 + \left(11 a^{5} + 10 a^{4} + a^{3} + 28 a^{2} + 31 a + 18\right)\cdot 43^{2} + \left(10 a^{5} + a^{3} + 20 a^{2} + 8 a + 11\right)\cdot 43^{3} + \left(10 a^{5} + 25 a^{4} + 18 a^{3} + 37 a^{2} + a + 24\right)\cdot 43^{4} + \left(2 a^{5} + 41 a^{3} + 7 a^{2} + 29 a + 6\right)\cdot 43^{5} + \left(15 a^{5} + 15 a^{4} + 6 a^{3} + 12 a^{2} + 10 a + 23\right)\cdot 43^{6} + \left(28 a^{5} + 36 a^{4} + 19 a^{3} + 12 a^{2} + 2 a + 39\right)\cdot 43^{7} + \left(22 a^{5} + 4 a^{4} + 13 a^{3} + 31 a^{2} + 42 a + 3\right)\cdot 43^{8} +O(43^{9})$$ 19*a^5 + 20*a^4 + 28*a^3 + 32*a^2 + 24*a + 16 + (10*a^5 + 41*a^4 + 26*a^3 + 26*a^2 + 6*a + 27)*43 + (11*a^5 + 10*a^4 + a^3 + 28*a^2 + 31*a + 18)*43^2 + (10*a^5 + a^3 + 20*a^2 + 8*a + 11)*43^3 + (10*a^5 + 25*a^4 + 18*a^3 + 37*a^2 + a + 24)*43^4 + (2*a^5 + 41*a^3 + 7*a^2 + 29*a + 6)*43^5 + (15*a^5 + 15*a^4 + 6*a^3 + 12*a^2 + 10*a + 23)*43^6 + (28*a^5 + 36*a^4 + 19*a^3 + 12*a^2 + 2*a + 39)*43^7 + (22*a^5 + 4*a^4 + 13*a^3 + 31*a^2 + 42*a + 3)*43^8+O(43^9) $r_{ 7 }$ $=$ $$17 a^{5} + 5 a^{4} + 38 a^{3} + 19 a^{2} + 26 a + 16 + \left(22 a^{5} + 11 a^{4} + 26 a^{3} + 19 a^{2} + 25 a + 21\right)\cdot 43 + \left(9 a^{5} + 31 a^{4} + 11 a^{3} + 2 a^{2} + 18 a + 5\right)\cdot 43^{2} + \left(5 a^{5} + 31 a^{4} + 28 a^{3} + 29 a^{2} + 41 a + 27\right)\cdot 43^{3} + \left(5 a^{5} + 38 a^{4} + 39 a^{3} + 41 a^{2} + 39 a + 21\right)\cdot 43^{4} + \left(9 a^{5} + a^{4} + 5 a^{3} + 16 a^{2} + 12 a + 34\right)\cdot 43^{5} + \left(11 a^{5} + 20 a^{4} + 25 a^{3} + 39 a^{2} + 19 a + 10\right)\cdot 43^{6} + \left(28 a^{5} + 8 a^{4} + 13 a^{3} + 15 a^{2} + 14 a + 5\right)\cdot 43^{7} + \left(42 a^{5} + 26 a^{4} + 24 a^{3} + 22 a^{2} + 35 a + 7\right)\cdot 43^{8} +O(43^{9})$$ 17*a^5 + 5*a^4 + 38*a^3 + 19*a^2 + 26*a + 16 + (22*a^5 + 11*a^4 + 26*a^3 + 19*a^2 + 25*a + 21)*43 + (9*a^5 + 31*a^4 + 11*a^3 + 2*a^2 + 18*a + 5)*43^2 + (5*a^5 + 31*a^4 + 28*a^3 + 29*a^2 + 41*a + 27)*43^3 + (5*a^5 + 38*a^4 + 39*a^3 + 41*a^2 + 39*a + 21)*43^4 + (9*a^5 + a^4 + 5*a^3 + 16*a^2 + 12*a + 34)*43^5 + (11*a^5 + 20*a^4 + 25*a^3 + 39*a^2 + 19*a + 10)*43^6 + (28*a^5 + 8*a^4 + 13*a^3 + 15*a^2 + 14*a + 5)*43^7 + (42*a^5 + 26*a^4 + 24*a^3 + 22*a^2 + 35*a + 7)*43^8+O(43^9) $r_{ 8 }$ $=$ $$24 a^{5} + 17 a^{4} + 18 a^{3} + 39 a^{2} + 8 a + 17 + \left(23 a^{5} + 40 a^{4} + 14 a^{3} + 7 a^{2} + a + 20\right)\cdot 43 + \left(22 a^{5} + 31 a^{4} + 3 a^{3} + 42 a^{2} + 25 a + 29\right)\cdot 43^{2} + \left(10 a^{5} + 18 a^{4} + 35 a^{3} + 7 a^{2} + 2 a + 3\right)\cdot 43^{3} + \left(9 a^{5} + 31 a^{4} + 31 a^{3} + 4 a^{2} + 7 a + 7\right)\cdot 43^{4} + \left(27 a^{4} + 19 a^{3} + 26 a^{2} + 29 a + 21\right)\cdot 43^{5} + \left(37 a^{5} + 41 a^{4} + 26 a^{3} + 38 a^{2} + 6 a + 7\right)\cdot 43^{6} + \left(19 a^{5} + 18 a^{4} + 29 a^{3} + 6 a^{2} + 24\right)\cdot 43^{7} + \left(30 a^{5} + 37 a^{4} + 38 a^{3} + 18 a^{2} + 13 a + 35\right)\cdot 43^{8} +O(43^{9})$$ 24*a^5 + 17*a^4 + 18*a^3 + 39*a^2 + 8*a + 17 + (23*a^5 + 40*a^4 + 14*a^3 + 7*a^2 + a + 20)*43 + (22*a^5 + 31*a^4 + 3*a^3 + 42*a^2 + 25*a + 29)*43^2 + (10*a^5 + 18*a^4 + 35*a^3 + 7*a^2 + 2*a + 3)*43^3 + (9*a^5 + 31*a^4 + 31*a^3 + 4*a^2 + 7*a + 7)*43^4 + (27*a^4 + 19*a^3 + 26*a^2 + 29*a + 21)*43^5 + (37*a^5 + 41*a^4 + 26*a^3 + 38*a^2 + 6*a + 7)*43^6 + (19*a^5 + 18*a^4 + 29*a^3 + 6*a^2 + 24)*43^7 + (30*a^5 + 37*a^4 + 38*a^3 + 18*a^2 + 13*a + 35)*43^8+O(43^9) $r_{ 9 }$ $=$ $$19 a^{5} + 35 a^{4} + 16 a^{3} + 10 a^{2} + 11 a + 5 + \left(39 a^{5} + 30 a^{4} + 19 a^{3} + 21 a^{2} + 21 a + 31\right)\cdot 43 + \left(28 a^{5} + 23 a^{4} + 34 a^{3} + 17 a^{2} + 16 a + 29\right)\cdot 43^{2} + \left(15 a^{5} + 2 a^{3} + 8 a^{2} + 27 a\right)\cdot 43^{3} + \left(22 a^{5} + 31 a^{4} + 16 a^{3} + 31 a^{2} + 9 a + 26\right)\cdot 43^{4} + \left(25 a^{5} + 2 a^{3} + 35 a^{2} + 29 a + 16\right)\cdot 43^{5} + \left(21 a^{5} + 12 a^{4} + 42 a^{3} + 40 a^{2} + 19 a + 42\right)\cdot 43^{6} + \left(36 a^{5} + 40 a^{4} + 22 a^{3} + 31 a^{2} + 28 a + 8\right)\cdot 43^{7} + \left(14 a^{5} + 21 a^{4} + 24 a^{3} + 24 a^{2} + 41 a + 3\right)\cdot 43^{8} +O(43^{9})$$ 19*a^5 + 35*a^4 + 16*a^3 + 10*a^2 + 11*a + 5 + (39*a^5 + 30*a^4 + 19*a^3 + 21*a^2 + 21*a + 31)*43 + (28*a^5 + 23*a^4 + 34*a^3 + 17*a^2 + 16*a + 29)*43^2 + (15*a^5 + 2*a^3 + 8*a^2 + 27*a)*43^3 + (22*a^5 + 31*a^4 + 16*a^3 + 31*a^2 + 9*a + 26)*43^4 + (25*a^5 + 2*a^3 + 35*a^2 + 29*a + 16)*43^5 + (21*a^5 + 12*a^4 + 42*a^3 + 40*a^2 + 19*a + 42)*43^6 + (36*a^5 + 40*a^4 + 22*a^3 + 31*a^2 + 28*a + 8)*43^7 + (14*a^5 + 21*a^4 + 24*a^3 + 24*a^2 + 41*a + 3)*43^8+O(43^9) $r_{ 10 }$ $=$ $$6 a^{5} + 26 a^{4} + 37 a^{3} + 6 a^{2} + 36 a + 29 + \left(10 a^{5} + 35 a^{4} + 9 a^{3} + 29 a^{2} + 26 a + 32\right)\cdot 43 + \left(35 a^{5} + 28 a^{4} + 31 a^{3} + 4 a^{2} + 20 a + 42\right)\cdot 43^{2} + \left(19 a^{5} + 31 a^{4} + 27 a^{3} + 35 a^{2} + 8 a\right)\cdot 43^{3} + \left(36 a^{5} + 14 a^{4} + 38 a^{3} + 40 a^{2} + 6 a\right)\cdot 43^{4} + \left(31 a^{5} + 16 a^{4} + 34 a^{3} + 18 a^{2} + 42 a + 26\right)\cdot 43^{5} + \left(27 a^{5} + 30 a^{4} + 33 a^{3} + 7 a + 34\right)\cdot 43^{6} + \left(15 a^{5} + 40 a^{4} + 17 a^{3} + 4 a^{2} + 37 a + 28\right)\cdot 43^{7} + \left(34 a^{5} + 26 a^{4} + 2 a^{3} + 33 a^{2} + 20 a + 7\right)\cdot 43^{8} +O(43^{9})$$ 6*a^5 + 26*a^4 + 37*a^3 + 6*a^2 + 36*a + 29 + (10*a^5 + 35*a^4 + 9*a^3 + 29*a^2 + 26*a + 32)*43 + (35*a^5 + 28*a^4 + 31*a^3 + 4*a^2 + 20*a + 42)*43^2 + (19*a^5 + 31*a^4 + 27*a^3 + 35*a^2 + 8*a)*43^3 + (36*a^5 + 14*a^4 + 38*a^3 + 40*a^2 + 6*a)*43^4 + (31*a^5 + 16*a^4 + 34*a^3 + 18*a^2 + 42*a + 26)*43^5 + (27*a^5 + 30*a^4 + 33*a^3 + 7*a + 34)*43^6 + (15*a^5 + 40*a^4 + 17*a^3 + 4*a^2 + 37*a + 28)*43^7 + (34*a^5 + 26*a^4 + 2*a^3 + 33*a^2 + 20*a + 7)*43^8+O(43^9) $r_{ 11 }$ $=$ $$38 a^{5} + 39 a^{4} + 4 a^{3} + 2 a^{2} + 5 a + 35 + \left(26 a^{5} + 14 a^{4} + 32 a^{3} + 37 a^{2} + 12 a + 14\right)\cdot 43 + \left(41 a^{5} + 3 a^{4} + 41 a^{3} + 16 a^{2} + 11 a + 24\right)\cdot 43^{2} + \left(40 a^{5} + 2 a^{4} + 28 a^{3} + 42 a^{2} + 29 a + 31\right)\cdot 43^{3} + \left(15 a^{5} + 31 a^{4} + 29 a^{3} + 21 a^{2} + a + 15\right)\cdot 43^{4} + \left(21 a^{5} + 30 a^{4} + 5 a^{3} + 36 a^{2} + 31 a + 21\right)\cdot 43^{5} + \left(9 a^{5} + 29 a^{4} + 20 a^{3} + a^{2} + 20 a + 27\right)\cdot 43^{6} + \left(15 a^{5} + 3 a^{4} + 18 a^{3} + 29 a^{2} + 21 a + 26\right)\cdot 43^{7} + \left(20 a^{5} + 24 a^{4} + 40 a^{3} + 38 a^{2} + 6 a + 12\right)\cdot 43^{8} +O(43^{9})$$ 38*a^5 + 39*a^4 + 4*a^3 + 2*a^2 + 5*a + 35 + (26*a^5 + 14*a^4 + 32*a^3 + 37*a^2 + 12*a + 14)*43 + (41*a^5 + 3*a^4 + 41*a^3 + 16*a^2 + 11*a + 24)*43^2 + (40*a^5 + 2*a^4 + 28*a^3 + 42*a^2 + 29*a + 31)*43^3 + (15*a^5 + 31*a^4 + 29*a^3 + 21*a^2 + a + 15)*43^4 + (21*a^5 + 30*a^4 + 5*a^3 + 36*a^2 + 31*a + 21)*43^5 + (9*a^5 + 29*a^4 + 20*a^3 + a^2 + 20*a + 27)*43^6 + (15*a^5 + 3*a^4 + 18*a^3 + 29*a^2 + 21*a + 26)*43^7 + (20*a^5 + 24*a^4 + 40*a^3 + 38*a^2 + 6*a + 12)*43^8+O(43^9) $r_{ 12 }$ $=$ $$6 a^{5} + 21 a^{4} + 26 a^{3} + 24 a^{2} + 39 a + 34 + \left(7 a^{5} + 5 a^{4} + 4 a^{3} + 36 a^{2} + 25 a + 13\right)\cdot 43 + \left(14 a^{5} + 13 a^{4} + 2 a^{3} + 24 a^{2} + 12 a + 26\right)\cdot 43^{2} + \left(37 a^{5} + 13 a^{4} + 16 a^{3} + 34 a^{2} + 12\right)\cdot 43^{3} + \left(15 a^{5} + 24 a^{4} + 6 a^{3} + 32 a^{2} + 36 a + 3\right)\cdot 43^{4} + \left(37 a^{5} + 3 a^{4} + 3 a^{3} + 11 a^{2} + 42 a + 12\right)\cdot 43^{5} + \left(21 a^{4} + a^{3} + 41 a^{2} + 34 a + 4\right)\cdot 43^{6} + \left(39 a^{5} + 25 a^{4} + 33 a^{3} + 18 a^{2} + 34 a + 30\right)\cdot 43^{7} + \left(31 a^{5} + 25 a^{4} + 17 a^{3} + 26 a^{2} + 14 a + 4\right)\cdot 43^{8} +O(43^{9})$$ 6*a^5 + 21*a^4 + 26*a^3 + 24*a^2 + 39*a + 34 + (7*a^5 + 5*a^4 + 4*a^3 + 36*a^2 + 25*a + 13)*43 + (14*a^5 + 13*a^4 + 2*a^3 + 24*a^2 + 12*a + 26)*43^2 + (37*a^5 + 13*a^4 + 16*a^3 + 34*a^2 + 12)*43^3 + (15*a^5 + 24*a^4 + 6*a^3 + 32*a^2 + 36*a + 3)*43^4 + (37*a^5 + 3*a^4 + 3*a^3 + 11*a^2 + 42*a + 12)*43^5 + (21*a^4 + a^3 + 41*a^2 + 34*a + 4)*43^6 + (39*a^5 + 25*a^4 + 33*a^3 + 18*a^2 + 34*a + 30)*43^7 + (31*a^5 + 25*a^4 + 17*a^3 + 26*a^2 + 14*a + 4)*43^8+O(43^9) $r_{ 13 }$ $=$ $$37 a^{5} + 7 a^{4} + 20 a^{3} + 16 a^{2} + 29 a + 36 + \left(2 a^{5} + 30 a^{4} + 4 a^{3} + 23 a^{2} + 15 a + 39\right)\cdot 43 + \left(25 a^{5} + 19 a^{4} + 25 a^{3} + 2 a^{2} + 22 a + 27\right)\cdot 43^{2} + \left(40 a^{5} + 19 a^{4} + 16 a^{3} + 25 a^{2} + 19 a + 24\right)\cdot 43^{3} + \left(14 a^{5} + 28 a^{4} + 16 a^{3} + 11 a^{2} + 13 a + 30\right)\cdot 43^{4} + \left(26 a^{5} + 22 a^{4} + 17 a^{3} + 21 a^{2} + 23 a + 2\right)\cdot 43^{5} + \left(31 a^{5} + 7 a^{4} + 23 a^{3} + 25 a^{2} + 11 a + 14\right)\cdot 43^{6} + \left(37 a^{5} + 11 a^{4} + 14 a^{2} + 7 a + 12\right)\cdot 43^{7} + \left(13 a^{5} + 20 a^{4} + 4 a^{3} + 31 a^{2} + 7 a + 32\right)\cdot 43^{8} +O(43^{9})$$ 37*a^5 + 7*a^4 + 20*a^3 + 16*a^2 + 29*a + 36 + (2*a^5 + 30*a^4 + 4*a^3 + 23*a^2 + 15*a + 39)*43 + (25*a^5 + 19*a^4 + 25*a^3 + 2*a^2 + 22*a + 27)*43^2 + (40*a^5 + 19*a^4 + 16*a^3 + 25*a^2 + 19*a + 24)*43^3 + (14*a^5 + 28*a^4 + 16*a^3 + 11*a^2 + 13*a + 30)*43^4 + (26*a^5 + 22*a^4 + 17*a^3 + 21*a^2 + 23*a + 2)*43^5 + (31*a^5 + 7*a^4 + 23*a^3 + 25*a^2 + 11*a + 14)*43^6 + (37*a^5 + 11*a^4 + 14*a^2 + 7*a + 12)*43^7 + (13*a^5 + 20*a^4 + 4*a^3 + 31*a^2 + 7*a + 32)*43^8+O(43^9) $r_{ 14 }$ $=$ $$7 a^{5} + 21 a^{4} + 10 a^{3} + 9 a^{2} + 5 a + 12 + \left(24 a^{5} + 41 a^{4} + 9 a^{3} + 9 a^{2} + 21 a + 9\right)\cdot 43 + \left(26 a^{5} + 6 a^{4} + 24 a^{3} + 5 a^{2} + 24 a + 18\right)\cdot 43^{2} + \left(12 a^{5} + 26 a^{4} + 8 a^{3} + 41 a^{2} + 33 a + 34\right)\cdot 43^{3} + \left(29 a^{5} + 5 a^{4} + 30 a^{3} + 14 a^{2} + 25 a + 3\right)\cdot 43^{4} + \left(7 a^{5} + 32 a^{4} + 19 a^{3} + 14 a^{2} + 5 a + 40\right)\cdot 43^{5} + \left(35 a^{5} + 4 a^{4} + 11 a^{3} + 35 a^{2} + 11 a + 32\right)\cdot 43^{6} + \left(7 a^{5} + 11 a^{4} + 27 a^{2} + 19 a + 25\right)\cdot 43^{7} + \left(5 a^{5} + 29 a^{4} + 39 a^{3} + 33 a^{2} + 15 a + 9\right)\cdot 43^{8} +O(43^{9})$$ 7*a^5 + 21*a^4 + 10*a^3 + 9*a^2 + 5*a + 12 + (24*a^5 + 41*a^4 + 9*a^3 + 9*a^2 + 21*a + 9)*43 + (26*a^5 + 6*a^4 + 24*a^3 + 5*a^2 + 24*a + 18)*43^2 + (12*a^5 + 26*a^4 + 8*a^3 + 41*a^2 + 33*a + 34)*43^3 + (29*a^5 + 5*a^4 + 30*a^3 + 14*a^2 + 25*a + 3)*43^4 + (7*a^5 + 32*a^4 + 19*a^3 + 14*a^2 + 5*a + 40)*43^5 + (35*a^5 + 4*a^4 + 11*a^3 + 35*a^2 + 11*a + 32)*43^6 + (7*a^5 + 11*a^4 + 27*a^2 + 19*a + 25)*43^7 + (5*a^5 + 29*a^4 + 39*a^3 + 33*a^2 + 15*a + 9)*43^8+O(43^9) $r_{ 15 }$ $=$ $$26 a^{5} + 38 a^{4} + 5 a^{3} + 24 a^{2} + 17 a + 3 + \left(20 a^{5} + 31 a^{4} + 16 a^{3} + 23 a^{2} + 17 a + 3\right)\cdot 43 + \left(33 a^{5} + 11 a^{4} + 31 a^{3} + 40 a^{2} + 24 a + 33\right)\cdot 43^{2} + \left(37 a^{5} + 11 a^{4} + 14 a^{3} + 13 a^{2} + a + 14\right)\cdot 43^{3} + \left(37 a^{5} + 4 a^{4} + 3 a^{3} + a^{2} + 3 a + 36\right)\cdot 43^{4} + \left(33 a^{5} + 41 a^{4} + 37 a^{3} + 26 a^{2} + 30 a + 4\right)\cdot 43^{5} + \left(31 a^{5} + 22 a^{4} + 17 a^{3} + 3 a^{2} + 23 a + 12\right)\cdot 43^{6} + \left(14 a^{5} + 34 a^{4} + 29 a^{3} + 27 a^{2} + 28 a + 4\right)\cdot 43^{7} + \left(16 a^{4} + 18 a^{3} + 20 a^{2} + 7 a + 41\right)\cdot 43^{8} +O(43^{9})$$ 26*a^5 + 38*a^4 + 5*a^3 + 24*a^2 + 17*a + 3 + (20*a^5 + 31*a^4 + 16*a^3 + 23*a^2 + 17*a + 3)*43 + (33*a^5 + 11*a^4 + 31*a^3 + 40*a^2 + 24*a + 33)*43^2 + (37*a^5 + 11*a^4 + 14*a^3 + 13*a^2 + a + 14)*43^3 + (37*a^5 + 4*a^4 + 3*a^3 + a^2 + 3*a + 36)*43^4 + (33*a^5 + 41*a^4 + 37*a^3 + 26*a^2 + 30*a + 4)*43^5 + (31*a^5 + 22*a^4 + 17*a^3 + 3*a^2 + 23*a + 12)*43^6 + (14*a^5 + 34*a^4 + 29*a^3 + 27*a^2 + 28*a + 4)*43^7 + (16*a^4 + 18*a^3 + 20*a^2 + 7*a + 41)*43^8+O(43^9) $r_{ 16 }$ $=$ $$14 a^{5} + 12 a^{4} + 31 a^{3} + 8 a^{2} + 38 a + 37 + \left(22 a^{5} + 22 a^{4} + 16 a^{3} + 25 a^{2} + 33 a + 40\right)\cdot 43 + \left(11 a^{5} + 30 a^{4} + a^{3} + 33 a^{2} + 4 a + 27\right)\cdot 43^{2} + \left(38 a^{5} + 8 a^{4} + 9 a^{3} + 33 a^{2} + 27 a + 18\right)\cdot 43^{3} + \left(15 a^{5} + 14 a^{4} + 21 a^{3} + 39 a^{2} + 30 a + 23\right)\cdot 43^{4} + \left(20 a^{5} + 19 a^{4} + 11 a^{3} + 25 a^{2} + 14 a + 2\right)\cdot 43^{5} + \left(32 a^{5} + 8 a^{4} + 21 a^{3} + 19 a^{2} + 27 a + 26\right)\cdot 43^{6} + \left(18 a^{5} + 30 a^{4} + 31 a^{3} + 3 a^{2} + 16 a + 31\right)\cdot 43^{7} + \left(9 a^{5} + 21 a^{4} + 31 a^{3} + 36 a^{2} + 12 a + 38\right)\cdot 43^{8} +O(43^{9})$$ 14*a^5 + 12*a^4 + 31*a^3 + 8*a^2 + 38*a + 37 + (22*a^5 + 22*a^4 + 16*a^3 + 25*a^2 + 33*a + 40)*43 + (11*a^5 + 30*a^4 + a^3 + 33*a^2 + 4*a + 27)*43^2 + (38*a^5 + 8*a^4 + 9*a^3 + 33*a^2 + 27*a + 18)*43^3 + (15*a^5 + 14*a^4 + 21*a^3 + 39*a^2 + 30*a + 23)*43^4 + (20*a^5 + 19*a^4 + 11*a^3 + 25*a^2 + 14*a + 2)*43^5 + (32*a^5 + 8*a^4 + 21*a^3 + 19*a^2 + 27*a + 26)*43^6 + (18*a^5 + 30*a^4 + 31*a^3 + 3*a^2 + 16*a + 31)*43^7 + (9*a^5 + 21*a^4 + 31*a^3 + 36*a^2 + 12*a + 38)*43^8+O(43^9)

## Generators of the action on the roots $r_1, \ldots, r_{ 16 }$

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.