# Properties

 Label 2.273.16t60.a.b Dimension $2$ Group $\SL(2,3):C_2$ Conductor $273$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: 16T60 Conductor: $$273$$$$\medspace = 3 \cdot 7 \cdot 13$$ Artin stem field: Galois closure of 16.0.30853268336830129281.2 Galois orbit size: $4$ Smallest permutation container: 16T60 Parity: odd Determinant: 1.273.6t1.d.a Projective image: $A_4$ Projective stem field: Galois closure of 4.0.74529.1

## Defining polynomial

 $f(x)$ $=$ $$x^{16} - 5 x^{15} + 31 x^{13} - 8 x^{12} - 101 x^{11} + 7 x^{10} + 183 x^{9} + 32 x^{8} - 183 x^{7} - 61 x^{6} + 112 x^{5} + 78 x^{4} - 9 x^{3} - 28 x^{2} - 7 x + 7$$ x^16 - 5*x^15 + 31*x^13 - 8*x^12 - 101*x^11 + 7*x^10 + 183*x^9 + 32*x^8 - 183*x^7 - 61*x^6 + 112*x^5 + 78*x^4 - 9*x^3 - 28*x^2 - 7*x + 7 .

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $$x^{6} + 35x^{3} + 4x^{2} + 30x + 2$$

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

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.