# Properties

 Label 2.3040.12t18.b.b Dimension $2$ Group $C_6\times S_3$ Conductor $3040$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $C_6\times S_3$ Conductor: $$3040$$$$\medspace = 2^{5} \cdot 5 \cdot 19$$ Artin stem field: Galois closure of 12.0.34162868224000000.2 Galois orbit size: $2$ Smallest permutation container: $C_6\times S_3$ Parity: odd Determinant: 1.760.6t1.b.a Projective image: $S_3$ Projective stem field: Galois closure of 3.1.14440.1

## Defining polynomial

 $f(x)$ $=$ $$x^{12} - 10x^{10} + 17x^{8} + 42x^{6} + 46x^{4} - 8x^{2} + 1$$ x^12 - 10*x^10 + 17*x^8 + 42*x^6 + 46*x^4 - 8*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:
 $r_{ 1 }$ $=$ $$10 a^{5} + 16 a^{4} + 2 a^{2} + 16 a + 11 + \left(13 a^{5} + 9 a^{4} + 14 a^{3} + 3 a^{2} + 3 a + 7\right)\cdot 17 + \left(11 a^{5} + 14 a^{4} + 5 a^{2} + 13 a + 6\right)\cdot 17^{2} + \left(14 a^{5} + 14 a^{4} + 11 a^{3} + a^{2} + 12 a + 5\right)\cdot 17^{3} + \left(16 a^{5} + 7 a^{4} + a^{3} + 9 a^{2} + 16 a + 9\right)\cdot 17^{4} + \left(10 a^{5} + 3 a^{3} + 11 a^{2} + 3\right)\cdot 17^{5} + \left(11 a^{5} + 10 a^{4} + 3 a^{3} + 8 a^{2} + 2 a + 16\right)\cdot 17^{6} + \left(9 a^{5} + 11 a^{4} + 13 a^{3} + a^{2} + 11 a + 4\right)\cdot 17^{7} + \left(11 a^{5} + 11 a^{4} + 16 a^{3} + 3 a^{2} + 8 a + 4\right)\cdot 17^{8} + \left(2 a^{5} + 4 a^{4} + 11 a^{3} + 9 a^{2} + 6 a + 2\right)\cdot 17^{9} +O(17^{10})$$ 10*a^5 + 16*a^4 + 2*a^2 + 16*a + 11 + (13*a^5 + 9*a^4 + 14*a^3 + 3*a^2 + 3*a + 7)*17 + (11*a^5 + 14*a^4 + 5*a^2 + 13*a + 6)*17^2 + (14*a^5 + 14*a^4 + 11*a^3 + a^2 + 12*a + 5)*17^3 + (16*a^5 + 7*a^4 + a^3 + 9*a^2 + 16*a + 9)*17^4 + (10*a^5 + 3*a^3 + 11*a^2 + 3)*17^5 + (11*a^5 + 10*a^4 + 3*a^3 + 8*a^2 + 2*a + 16)*17^6 + (9*a^5 + 11*a^4 + 13*a^3 + a^2 + 11*a + 4)*17^7 + (11*a^5 + 11*a^4 + 16*a^3 + 3*a^2 + 8*a + 4)*17^8 + (2*a^5 + 4*a^4 + 11*a^3 + 9*a^2 + 6*a + 2)*17^9+O(17^10) $r_{ 2 }$ $=$ $$14 a^{5} + 13 a^{4} + 4 a^{3} + 14 a^{2} + 13 a + 16 + \left(2 a^{5} + 2 a^{4} + 15 a^{3} + 11 a^{2} + 7 a + 11\right)\cdot 17 + \left(5 a^{5} + 10 a^{4} + 6 a^{3} + 14 a^{2} + 2 a + 2\right)\cdot 17^{2} + \left(5 a^{5} + a^{4} + 16 a^{3} + 14 a + 9\right)\cdot 17^{3} + \left(a^{5} + 13 a^{4} + 5 a^{3} + 8 a^{2} + 9 a + 3\right)\cdot 17^{4} + \left(5 a^{5} + 12 a^{4} + 4 a^{2} + a + 10\right)\cdot 17^{5} + \left(12 a^{5} + 4 a^{4} + a^{3} + 2 a^{2} + 15 a + 13\right)\cdot 17^{6} + \left(a^{5} + 6 a^{4} + 2 a^{3} + 7 a + 3\right)\cdot 17^{7} + \left(7 a^{4} + 15 a^{3} + 4 a^{2} + 14 a + 12\right)\cdot 17^{8} + \left(13 a^{5} + 14 a^{4} + 5 a^{3} + 10 a^{2} + 2 a + 1\right)\cdot 17^{9} +O(17^{10})$$ 14*a^5 + 13*a^4 + 4*a^3 + 14*a^2 + 13*a + 16 + (2*a^5 + 2*a^4 + 15*a^3 + 11*a^2 + 7*a + 11)*17 + (5*a^5 + 10*a^4 + 6*a^3 + 14*a^2 + 2*a + 2)*17^2 + (5*a^5 + a^4 + 16*a^3 + 14*a + 9)*17^3 + (a^5 + 13*a^4 + 5*a^3 + 8*a^2 + 9*a + 3)*17^4 + (5*a^5 + 12*a^4 + 4*a^2 + a + 10)*17^5 + (12*a^5 + 4*a^4 + a^3 + 2*a^2 + 15*a + 13)*17^6 + (a^5 + 6*a^4 + 2*a^3 + 7*a + 3)*17^7 + (7*a^4 + 15*a^3 + 4*a^2 + 14*a + 12)*17^8 + (13*a^5 + 14*a^4 + 5*a^3 + 10*a^2 + 2*a + 1)*17^9+O(17^10) $r_{ 3 }$ $=$ $$6 a^{5} + 16 a^{4} + 8 a^{3} + 5 a^{2} + 15 a + 3 + \left(8 a^{4} + 3 a^{3} + 11 a^{2} + 14 a\right)\cdot 17 + \left(16 a^{5} + 11 a^{4} + 5 a^{3} + 11 a^{2} + 12 a + 5\right)\cdot 17^{2} + \left(a^{5} + 11 a^{4} + 15 a^{3} + a^{2} + 15 a + 16\right)\cdot 17^{3} + \left(10 a^{5} + 12 a^{4} + 2 a^{3} + 13 a^{2} + 13 a\right)\cdot 17^{4} + \left(3 a^{5} + 3 a^{4} + 10 a^{3} + 7 a^{2} + 9 a\right)\cdot 17^{5} + \left(6 a^{5} + 4 a^{4} + 12 a^{3} + 2 a^{2} + 14 a + 10\right)\cdot 17^{6} + \left(a^{5} + 7 a^{3} + 16 a^{2} + 16 a + 15\right)\cdot 17^{7} + \left(11 a^{5} + 15 a^{4} + 4 a^{2} + 16 a + 7\right)\cdot 17^{8} + \left(12 a^{5} + 5 a^{4} + 8 a^{3} + 4 a^{2} + 13 a + 10\right)\cdot 17^{9} +O(17^{10})$$ 6*a^5 + 16*a^4 + 8*a^3 + 5*a^2 + 15*a + 3 + (8*a^4 + 3*a^3 + 11*a^2 + 14*a)*17 + (16*a^5 + 11*a^4 + 5*a^3 + 11*a^2 + 12*a + 5)*17^2 + (a^5 + 11*a^4 + 15*a^3 + a^2 + 15*a + 16)*17^3 + (10*a^5 + 12*a^4 + 2*a^3 + 13*a^2 + 13*a)*17^4 + (3*a^5 + 3*a^4 + 10*a^3 + 7*a^2 + 9*a)*17^5 + (6*a^5 + 4*a^4 + 12*a^3 + 2*a^2 + 14*a + 10)*17^6 + (a^5 + 7*a^3 + 16*a^2 + 16*a + 15)*17^7 + (11*a^5 + 15*a^4 + 4*a^2 + 16*a + 7)*17^8 + (12*a^5 + 5*a^4 + 8*a^3 + 4*a^2 + 13*a + 10)*17^9+O(17^10) $r_{ 4 }$ $=$ $$15 a^{5} + 7 a^{4} + 2 a^{3} + 2 a^{2} + 12 a + 4 + \left(11 a^{5} + a^{4} + 4 a^{3} + 5 a^{2} + 12 a\right)\cdot 17 + \left(8 a^{5} + 4 a^{4} + 8 a^{3} + 13 a^{2} + 9 a + 15\right)\cdot 17^{2} + \left(11 a^{5} + 14 a^{4} + 6 a^{3} + 13 a^{2} + 3 a + 12\right)\cdot 17^{3} + \left(9 a^{5} + 12 a^{4} + 16 a^{3} + 14 a + 9\right)\cdot 17^{4} + \left(8 a^{5} + 10 a^{4} + a^{3} + 11 a + 16\right)\cdot 17^{5} + \left(7 a^{5} + 9 a^{4} + 2 a^{3} + 7 a^{2} + 5 a + 10\right)\cdot 17^{6} + \left(11 a^{5} + 9 a^{4} + 8 a^{3} + 11 a^{2} + 11 a + 10\right)\cdot 17^{7} + \left(14 a^{4} + 4 a^{3} + a^{2} + 11 a + 13\right)\cdot 17^{8} + \left(8 a^{5} + 15 a^{4} + 3 a^{3} + 13 a^{2} + 10 a + 15\right)\cdot 17^{9} +O(17^{10})$$ 15*a^5 + 7*a^4 + 2*a^3 + 2*a^2 + 12*a + 4 + (11*a^5 + a^4 + 4*a^3 + 5*a^2 + 12*a)*17 + (8*a^5 + 4*a^4 + 8*a^3 + 13*a^2 + 9*a + 15)*17^2 + (11*a^5 + 14*a^4 + 6*a^3 + 13*a^2 + 3*a + 12)*17^3 + (9*a^5 + 12*a^4 + 16*a^3 + 14*a + 9)*17^4 + (8*a^5 + 10*a^4 + a^3 + 11*a + 16)*17^5 + (7*a^5 + 9*a^4 + 2*a^3 + 7*a^2 + 5*a + 10)*17^6 + (11*a^5 + 9*a^4 + 8*a^3 + 11*a^2 + 11*a + 10)*17^7 + (14*a^4 + 4*a^3 + a^2 + 11*a + 13)*17^8 + (8*a^5 + 15*a^4 + 3*a^3 + 13*a^2 + 10*a + 15)*17^9+O(17^10) $r_{ 5 }$ $=$ $$11 a^{5} + 14 a^{4} + 11 a^{3} + 2 a^{2} + 15 a + \left(16 a^{5} + 6 a^{4} + 12 a^{3} + 13 a^{2} + 15 a + 11\right)\cdot 17 + \left(14 a^{5} + 11 a^{4} + 15 a^{3} + 4 a^{2} + 16 a + 5\right)\cdot 17^{2} + \left(8 a^{5} + 7 a^{4} + a^{3} + 3 a^{2} + 14 a + 13\right)\cdot 17^{3} + \left(9 a^{5} + 10 a^{4} + 5 a^{3} + 4 a^{2} + a + 12\right)\cdot 17^{4} + \left(a^{5} + 16 a^{4} + 7 a^{3} + 2 a^{2} + 16 a + 11\right)\cdot 17^{5} + \left(15 a^{5} + 13 a^{4} + 10 a^{3} + 10 a^{2} + 7 a + 15\right)\cdot 17^{6} + \left(6 a^{5} + 10 a^{4} + 6 a^{3} + 11 a^{2} + 16 a + 11\right)\cdot 17^{7} + \left(16 a^{5} + 16 a^{4} + 7 a^{3} + 16 a^{2} + 5 a + 3\right)\cdot 17^{8} + \left(7 a^{5} + 9 a^{3} + 7 a^{2} + 5 a + 6\right)\cdot 17^{9} +O(17^{10})$$ 11*a^5 + 14*a^4 + 11*a^3 + 2*a^2 + 15*a + (16*a^5 + 6*a^4 + 12*a^3 + 13*a^2 + 15*a + 11)*17 + (14*a^5 + 11*a^4 + 15*a^3 + 4*a^2 + 16*a + 5)*17^2 + (8*a^5 + 7*a^4 + a^3 + 3*a^2 + 14*a + 13)*17^3 + (9*a^5 + 10*a^4 + 5*a^3 + 4*a^2 + a + 12)*17^4 + (a^5 + 16*a^4 + 7*a^3 + 2*a^2 + 16*a + 11)*17^5 + (15*a^5 + 13*a^4 + 10*a^3 + 10*a^2 + 7*a + 15)*17^6 + (6*a^5 + 10*a^4 + 6*a^3 + 11*a^2 + 16*a + 11)*17^7 + (16*a^5 + 16*a^4 + 7*a^3 + 16*a^2 + 5*a + 3)*17^8 + (7*a^5 + 9*a^3 + 7*a^2 + 5*a + 6)*17^9+O(17^10) $r_{ 6 }$ $=$ $$15 a^{5} + 9 a^{4} + 13 a^{3} + 10 a^{2} + 4 a + 3 + \left(15 a^{5} + 4 a^{4} + 10 a^{3} + 10 a^{2} + 14 a + 2\right)\cdot 17 + \left(11 a^{5} + 6 a^{4} + 6 a^{3} + 10 a^{2} + 13 a + 13\right)\cdot 17^{2} + \left(8 a^{5} + a^{4} + 5 a^{3} + 8 a^{2} + 8 a + 1\right)\cdot 17^{3} + \left(8 a^{5} + 5 a^{4} + 4 a^{3} + 8 a + 2\right)\cdot 17^{4} + \left(2 a^{5} + 14 a^{4} + a^{3} + 13 a + 6\right)\cdot 17^{5} + \left(13 a^{5} + 13 a^{4} + 6 a^{3} + 12 a^{2} + 3 a + 8\right)\cdot 17^{6} + \left(4 a^{5} + 6 a^{4} + 8 a\right)\cdot 17^{7} + \left(4 a^{5} + 4 a^{4} + 5 a^{3} + 2 a^{2} + 5 a + 14\right)\cdot 17^{8} + \left(2 a^{5} + 15 a^{4} + 3 a^{3} + 15 a^{2} + 12 a + 10\right)\cdot 17^{9} +O(17^{10})$$ 15*a^5 + 9*a^4 + 13*a^3 + 10*a^2 + 4*a + 3 + (15*a^5 + 4*a^4 + 10*a^3 + 10*a^2 + 14*a + 2)*17 + (11*a^5 + 6*a^4 + 6*a^3 + 10*a^2 + 13*a + 13)*17^2 + (8*a^5 + a^4 + 5*a^3 + 8*a^2 + 8*a + 1)*17^3 + (8*a^5 + 5*a^4 + 4*a^3 + 8*a + 2)*17^4 + (2*a^5 + 14*a^4 + a^3 + 13*a + 6)*17^5 + (13*a^5 + 13*a^4 + 6*a^3 + 12*a^2 + 3*a + 8)*17^6 + (4*a^5 + 6*a^4 + 8*a)*17^7 + (4*a^5 + 4*a^4 + 5*a^3 + 2*a^2 + 5*a + 14)*17^8 + (2*a^5 + 15*a^4 + 3*a^3 + 15*a^2 + 12*a + 10)*17^9+O(17^10) $r_{ 7 }$ $=$ $$7 a^{5} + a^{4} + 15 a^{2} + a + 6 + \left(3 a^{5} + 7 a^{4} + 3 a^{3} + 13 a^{2} + 13 a + 9\right)\cdot 17 + \left(5 a^{5} + 2 a^{4} + 16 a^{3} + 11 a^{2} + 3 a + 10\right)\cdot 17^{2} + \left(2 a^{5} + 2 a^{4} + 5 a^{3} + 15 a^{2} + 4 a + 11\right)\cdot 17^{3} + \left(9 a^{4} + 15 a^{3} + 7 a^{2} + 7\right)\cdot 17^{4} + \left(6 a^{5} + 16 a^{4} + 13 a^{3} + 5 a^{2} + 16 a + 13\right)\cdot 17^{5} + \left(5 a^{5} + 6 a^{4} + 13 a^{3} + 8 a^{2} + 14 a\right)\cdot 17^{6} + \left(7 a^{5} + 5 a^{4} + 3 a^{3} + 15 a^{2} + 5 a + 12\right)\cdot 17^{7} + \left(5 a^{5} + 5 a^{4} + 13 a^{2} + 8 a + 12\right)\cdot 17^{8} + \left(14 a^{5} + 12 a^{4} + 5 a^{3} + 7 a^{2} + 10 a + 14\right)\cdot 17^{9} +O(17^{10})$$ 7*a^5 + a^4 + 15*a^2 + a + 6 + (3*a^5 + 7*a^4 + 3*a^3 + 13*a^2 + 13*a + 9)*17 + (5*a^5 + 2*a^4 + 16*a^3 + 11*a^2 + 3*a + 10)*17^2 + (2*a^5 + 2*a^4 + 5*a^3 + 15*a^2 + 4*a + 11)*17^3 + (9*a^4 + 15*a^3 + 7*a^2 + 7)*17^4 + (6*a^5 + 16*a^4 + 13*a^3 + 5*a^2 + 16*a + 13)*17^5 + (5*a^5 + 6*a^4 + 13*a^3 + 8*a^2 + 14*a)*17^6 + (7*a^5 + 5*a^4 + 3*a^3 + 15*a^2 + 5*a + 12)*17^7 + (5*a^5 + 5*a^4 + 13*a^2 + 8*a + 12)*17^8 + (14*a^5 + 12*a^4 + 5*a^3 + 7*a^2 + 10*a + 14)*17^9+O(17^10) $r_{ 8 }$ $=$ $$3 a^{5} + 4 a^{4} + 13 a^{3} + 3 a^{2} + 4 a + 1 + \left(14 a^{5} + 14 a^{4} + a^{3} + 5 a^{2} + 9 a + 5\right)\cdot 17 + \left(11 a^{5} + 6 a^{4} + 10 a^{3} + 2 a^{2} + 14 a + 14\right)\cdot 17^{2} + \left(11 a^{5} + 15 a^{4} + 16 a^{2} + 2 a + 7\right)\cdot 17^{3} + \left(15 a^{5} + 3 a^{4} + 11 a^{3} + 8 a^{2} + 7 a + 13\right)\cdot 17^{4} + \left(11 a^{5} + 4 a^{4} + 16 a^{3} + 12 a^{2} + 15 a + 6\right)\cdot 17^{5} + \left(4 a^{5} + 12 a^{4} + 15 a^{3} + 14 a^{2} + a + 3\right)\cdot 17^{6} + \left(15 a^{5} + 10 a^{4} + 14 a^{3} + 16 a^{2} + 9 a + 13\right)\cdot 17^{7} + \left(16 a^{5} + 9 a^{4} + a^{3} + 12 a^{2} + 2 a + 4\right)\cdot 17^{8} + \left(3 a^{5} + 2 a^{4} + 11 a^{3} + 6 a^{2} + 14 a + 15\right)\cdot 17^{9} +O(17^{10})$$ 3*a^5 + 4*a^4 + 13*a^3 + 3*a^2 + 4*a + 1 + (14*a^5 + 14*a^4 + a^3 + 5*a^2 + 9*a + 5)*17 + (11*a^5 + 6*a^4 + 10*a^3 + 2*a^2 + 14*a + 14)*17^2 + (11*a^5 + 15*a^4 + 16*a^2 + 2*a + 7)*17^3 + (15*a^5 + 3*a^4 + 11*a^3 + 8*a^2 + 7*a + 13)*17^4 + (11*a^5 + 4*a^4 + 16*a^3 + 12*a^2 + 15*a + 6)*17^5 + (4*a^5 + 12*a^4 + 15*a^3 + 14*a^2 + a + 3)*17^6 + (15*a^5 + 10*a^4 + 14*a^3 + 16*a^2 + 9*a + 13)*17^7 + (16*a^5 + 9*a^4 + a^3 + 12*a^2 + 2*a + 4)*17^8 + (3*a^5 + 2*a^4 + 11*a^3 + 6*a^2 + 14*a + 15)*17^9+O(17^10) $r_{ 9 }$ $=$ $$2 a^{5} + 10 a^{4} + 15 a^{3} + 15 a^{2} + 5 a + 13 + \left(5 a^{5} + 15 a^{4} + 12 a^{3} + 11 a^{2} + 4 a + 16\right)\cdot 17 + \left(8 a^{5} + 12 a^{4} + 8 a^{3} + 3 a^{2} + 7 a + 1\right)\cdot 17^{2} + \left(5 a^{5} + 2 a^{4} + 10 a^{3} + 3 a^{2} + 13 a + 4\right)\cdot 17^{3} + \left(7 a^{5} + 4 a^{4} + 16 a^{2} + 2 a + 7\right)\cdot 17^{4} + \left(8 a^{5} + 6 a^{4} + 15 a^{3} + 16 a^{2} + 5 a\right)\cdot 17^{5} + \left(9 a^{5} + 7 a^{4} + 14 a^{3} + 9 a^{2} + 11 a + 6\right)\cdot 17^{6} + \left(5 a^{5} + 7 a^{4} + 8 a^{3} + 5 a^{2} + 5 a + 6\right)\cdot 17^{7} + \left(16 a^{5} + 2 a^{4} + 12 a^{3} + 15 a^{2} + 5 a + 3\right)\cdot 17^{8} + \left(8 a^{5} + a^{4} + 13 a^{3} + 3 a^{2} + 6 a + 1\right)\cdot 17^{9} +O(17^{10})$$ 2*a^5 + 10*a^4 + 15*a^3 + 15*a^2 + 5*a + 13 + (5*a^5 + 15*a^4 + 12*a^3 + 11*a^2 + 4*a + 16)*17 + (8*a^5 + 12*a^4 + 8*a^3 + 3*a^2 + 7*a + 1)*17^2 + (5*a^5 + 2*a^4 + 10*a^3 + 3*a^2 + 13*a + 4)*17^3 + (7*a^5 + 4*a^4 + 16*a^2 + 2*a + 7)*17^4 + (8*a^5 + 6*a^4 + 15*a^3 + 16*a^2 + 5*a)*17^5 + (9*a^5 + 7*a^4 + 14*a^3 + 9*a^2 + 11*a + 6)*17^6 + (5*a^5 + 7*a^4 + 8*a^3 + 5*a^2 + 5*a + 6)*17^7 + (16*a^5 + 2*a^4 + 12*a^3 + 15*a^2 + 5*a + 3)*17^8 + (8*a^5 + a^4 + 13*a^3 + 3*a^2 + 6*a + 1)*17^9+O(17^10) $r_{ 10 }$ $=$ $$11 a^{5} + a^{4} + 9 a^{3} + 12 a^{2} + 2 a + 14 + \left(16 a^{5} + 8 a^{4} + 13 a^{3} + 5 a^{2} + 2 a + 16\right)\cdot 17 + \left(5 a^{4} + 11 a^{3} + 5 a^{2} + 4 a + 11\right)\cdot 17^{2} + \left(15 a^{5} + 5 a^{4} + a^{3} + 15 a^{2} + a\right)\cdot 17^{3} + \left(6 a^{5} + 4 a^{4} + 14 a^{3} + 3 a^{2} + 3 a + 16\right)\cdot 17^{4} + \left(13 a^{5} + 13 a^{4} + 6 a^{3} + 9 a^{2} + 7 a + 16\right)\cdot 17^{5} + \left(10 a^{5} + 12 a^{4} + 4 a^{3} + 14 a^{2} + 2 a + 6\right)\cdot 17^{6} + \left(15 a^{5} + 16 a^{4} + 9 a^{3} + 1\right)\cdot 17^{7} + \left(5 a^{5} + a^{4} + 16 a^{3} + 12 a^{2} + 9\right)\cdot 17^{8} + \left(4 a^{5} + 11 a^{4} + 8 a^{3} + 12 a^{2} + 3 a + 6\right)\cdot 17^{9} +O(17^{10})$$ 11*a^5 + a^4 + 9*a^3 + 12*a^2 + 2*a + 14 + (16*a^5 + 8*a^4 + 13*a^3 + 5*a^2 + 2*a + 16)*17 + (5*a^4 + 11*a^3 + 5*a^2 + 4*a + 11)*17^2 + (15*a^5 + 5*a^4 + a^3 + 15*a^2 + a)*17^3 + (6*a^5 + 4*a^4 + 14*a^3 + 3*a^2 + 3*a + 16)*17^4 + (13*a^5 + 13*a^4 + 6*a^3 + 9*a^2 + 7*a + 16)*17^5 + (10*a^5 + 12*a^4 + 4*a^3 + 14*a^2 + 2*a + 6)*17^6 + (15*a^5 + 16*a^4 + 9*a^3 + 1)*17^7 + (5*a^5 + a^4 + 16*a^3 + 12*a^2 + 9)*17^8 + (4*a^5 + 11*a^4 + 8*a^3 + 12*a^2 + 3*a + 6)*17^9+O(17^10) $r_{ 11 }$ $=$ $$6 a^{5} + 3 a^{4} + 6 a^{3} + 15 a^{2} + 2 a + \left(10 a^{4} + 4 a^{3} + 3 a^{2} + a + 6\right)\cdot 17 + \left(2 a^{5} + 5 a^{4} + a^{3} + 12 a^{2} + 11\right)\cdot 17^{2} + \left(8 a^{5} + 9 a^{4} + 15 a^{3} + 13 a^{2} + 2 a + 3\right)\cdot 17^{3} + \left(7 a^{5} + 6 a^{4} + 11 a^{3} + 12 a^{2} + 15 a + 4\right)\cdot 17^{4} + \left(15 a^{5} + 9 a^{3} + 14 a^{2} + 5\right)\cdot 17^{5} + \left(a^{5} + 3 a^{4} + 6 a^{3} + 6 a^{2} + 9 a + 1\right)\cdot 17^{6} + \left(10 a^{5} + 6 a^{4} + 10 a^{3} + 5 a^{2} + 5\right)\cdot 17^{7} + \left(9 a^{3} + 11 a + 13\right)\cdot 17^{8} + \left(9 a^{5} + 16 a^{4} + 7 a^{3} + 9 a^{2} + 11 a + 10\right)\cdot 17^{9} +O(17^{10})$$ 6*a^5 + 3*a^4 + 6*a^3 + 15*a^2 + 2*a + (10*a^4 + 4*a^3 + 3*a^2 + a + 6)*17 + (2*a^5 + 5*a^4 + a^3 + 12*a^2 + 11)*17^2 + (8*a^5 + 9*a^4 + 15*a^3 + 13*a^2 + 2*a + 3)*17^3 + (7*a^5 + 6*a^4 + 11*a^3 + 12*a^2 + 15*a + 4)*17^4 + (15*a^5 + 9*a^3 + 14*a^2 + 5)*17^5 + (a^5 + 3*a^4 + 6*a^3 + 6*a^2 + 9*a + 1)*17^6 + (10*a^5 + 6*a^4 + 10*a^3 + 5*a^2 + 5)*17^7 + (9*a^3 + 11*a + 13)*17^8 + (9*a^5 + 16*a^4 + 7*a^3 + 9*a^2 + 11*a + 10)*17^9+O(17^10) $r_{ 12 }$ $=$ $$2 a^{5} + 8 a^{4} + 4 a^{3} + 7 a^{2} + 13 a + 14 + \left(a^{5} + 12 a^{4} + 6 a^{3} + 6 a^{2} + 2 a + 14\right)\cdot 17 + \left(5 a^{5} + 10 a^{4} + 10 a^{3} + 6 a^{2} + 3 a + 3\right)\cdot 17^{2} + \left(8 a^{5} + 15 a^{4} + 11 a^{3} + 8 a^{2} + 8 a + 15\right)\cdot 17^{3} + \left(8 a^{5} + 11 a^{4} + 12 a^{3} + 16 a^{2} + 8 a + 14\right)\cdot 17^{4} + \left(14 a^{5} + 2 a^{4} + 15 a^{3} + 16 a^{2} + 3 a + 10\right)\cdot 17^{5} + \left(3 a^{5} + 3 a^{4} + 10 a^{3} + 4 a^{2} + 13 a + 8\right)\cdot 17^{6} + \left(12 a^{5} + 10 a^{4} + 16 a^{3} + 16 a^{2} + 8 a + 16\right)\cdot 17^{7} + \left(12 a^{5} + 12 a^{4} + 11 a^{3} + 14 a^{2} + 11 a + 2\right)\cdot 17^{8} + \left(14 a^{5} + a^{4} + 13 a^{3} + a^{2} + 4 a + 6\right)\cdot 17^{9} +O(17^{10})$$ 2*a^5 + 8*a^4 + 4*a^3 + 7*a^2 + 13*a + 14 + (a^5 + 12*a^4 + 6*a^3 + 6*a^2 + 2*a + 14)*17 + (5*a^5 + 10*a^4 + 10*a^3 + 6*a^2 + 3*a + 3)*17^2 + (8*a^5 + 15*a^4 + 11*a^3 + 8*a^2 + 8*a + 15)*17^3 + (8*a^5 + 11*a^4 + 12*a^3 + 16*a^2 + 8*a + 14)*17^4 + (14*a^5 + 2*a^4 + 15*a^3 + 16*a^2 + 3*a + 10)*17^5 + (3*a^5 + 3*a^4 + 10*a^3 + 4*a^2 + 13*a + 8)*17^6 + (12*a^5 + 10*a^4 + 16*a^3 + 16*a^2 + 8*a + 16)*17^7 + (12*a^5 + 12*a^4 + 11*a^3 + 14*a^2 + 11*a + 2)*17^8 + (14*a^5 + a^4 + 13*a^3 + a^2 + 4*a + 6)*17^9+O(17^10)

## Generators of the action on the roots $r_1, \ldots, r_{ 12 }$

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.