# Properties

 Label 2.448.16t60.a.d Dimension $2$ Group $\SL(2,3):C_2$ Conductor $448$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: 16T60 Conductor: $$448$$$$\medspace = 2^{6} \cdot 7$$ Artin stem field: Galois closure of 16.0.1584616432828678144.8 Galois orbit size: $4$ Smallest permutation container: 16T60 Parity: odd Determinant: 1.28.6t1.a.b Projective image: $A_4$ Projective stem field: Galois closure of 4.0.3136.1

## Defining polynomial

 $f(x)$ $=$ $$x^{16} - 4x^{14} + 4x^{12} - 4x^{10} + 10x^{8} + 4x^{6} + 4x^{4} + 4x^{2} + 1$$ x^16 - 4*x^14 + 4*x^12 - 4*x^10 + 10*x^8 + 4*x^6 + 4*x^4 + 4*x^2 + 1 .

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $$x^{6} + 2x^{4} + 10x^{2} + 3x + 3$$

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

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.