# Properties

 Label 2.124.16t60.a Dimension $2$ Group $\SL(2,3):C_2$ Conductor $124$ Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: 16T60 Conductor: $$124$$$$\medspace = 2^{2} \cdot 31$$ Artin number field: Galois closure of 16.0.55895067029733376.1 Galois orbit size: $4$ Smallest permutation container: 16T60 Parity: odd Projective image: $A_4$ Projective field: Galois closure of 4.0.15376.1

## Galois action

### Roots of defining polynomial

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

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 16 }$ Character values $c1$ $c2$ $c3$ $c4$ $1$ $1$ $()$ $2$ $2$ $2$ $2$ $1$ $2$ $(1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)$ $-2$ $-2$ $-2$ $-2$ $6$ $2$ $(1,6)(2,5)(3,15)(4,8)(7,11)(9,14)(10,13)(12,16)$ $0$ $0$ $0$ $0$ $4$ $3$ $(3,5,12)(4,11,13)(6,15,8)(7,16,14)$ $\zeta_{12}^{2}$ $-\zeta_{12}^{2} + 1$ $-\zeta_{12}^{2} + 1$ $\zeta_{12}^{2}$ $4$ $3$ $(3,12,5)(4,13,11)(6,8,15)(7,14,16)$ $-\zeta_{12}^{2} + 1$ $\zeta_{12}^{2}$ $\zeta_{12}^{2}$ $-\zeta_{12}^{2} + 1$ $1$ $4$ $(1,2,9,10)(3,16,11,8)(4,15,12,7)(5,14,13,6)$ $2 \zeta_{12}^{3}$ $-2 \zeta_{12}^{3}$ $2 \zeta_{12}^{3}$ $-2 \zeta_{12}^{3}$ $1$ $4$ $(1,10,9,2)(3,8,11,16)(4,7,12,15)(5,6,13,14)$ $-2 \zeta_{12}^{3}$ $2 \zeta_{12}^{3}$ $-2 \zeta_{12}^{3}$ $2 \zeta_{12}^{3}$ $6$ $4$ $(1,13,9,5)(2,6,10,14)(3,4,11,12)(7,16,15,8)$ $0$ $0$ $0$ $0$ $4$ $6$ $(1,9)(2,10)(3,4,5,11,12,13)(6,16,15,14,8,7)$ $\zeta_{12}^{2} - 1$ $-\zeta_{12}^{2}$ $-\zeta_{12}^{2}$ $\zeta_{12}^{2} - 1$ $4$ $6$ $(1,9)(2,10)(3,13,12,11,5,4)(6,7,8,14,15,16)$ $-\zeta_{12}^{2}$ $\zeta_{12}^{2} - 1$ $\zeta_{12}^{2} - 1$ $-\zeta_{12}^{2}$ $4$ $12$ $(1,2,9,10)(3,14,4,8,5,7,11,6,12,16,13,15)$ $\zeta_{12}^{3} - \zeta_{12}$ $-\zeta_{12}$ $\zeta_{12}$ $-\zeta_{12}^{3} + \zeta_{12}$ $4$ $12$ $(1,2,9,10)(3,7,13,8,12,14,11,15,5,16,4,6)$ $\zeta_{12}$ $-\zeta_{12}^{3} + \zeta_{12}$ $\zeta_{12}^{3} - \zeta_{12}$ $-\zeta_{12}$ $4$ $12$ $(1,10,9,2)(3,6,4,16,5,15,11,14,12,8,13,7)$ $-\zeta_{12}^{3} + \zeta_{12}$ $\zeta_{12}$ $-\zeta_{12}$ $\zeta_{12}^{3} - \zeta_{12}$ $4$ $12$ $(1,10,9,2)(3,15,13,16,12,6,11,7,5,8,4,14)$ $-\zeta_{12}$ $\zeta_{12}^{3} - \zeta_{12}$ $-\zeta_{12}^{3} + \zeta_{12}$ $\zeta_{12}$
The blue line marks the conjugacy class containing complex conjugation.