# Properties

 Label 2.2888.12t18.f Dimension $2$ Group $C_6\times S_3$ Conductor $2888$ Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $C_6\times S_3$ Conductor: $$2888$$$$\medspace = 2^{3} \cdot 19^{2}$$ Artin number field: Galois closure of 12.0.4452139149819904.6 Galois orbit size: $2$ Smallest permutation container: $C_6\times S_3$ Parity: odd Projective image: $S_3$ Projective field: Galois closure of 3.1.2888.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $$x^{6} + 10x^{3} + 11x^{2} + 11x + 2$$
Roots:
 $r_{ 1 }$ $=$ $$3 a^{5} + 3 a^{4} + 10 a^{3} + 5 a^{2} + 11 a + 1 + \left(8 a^{5} + a^{4} + 2 a^{3} + 7 a^{2} + 7 a + 8\right)\cdot 13 + \left(7 a^{5} + 12 a^{4} + 5 a^{3} + 2 a^{2} + 12 a + 9\right)\cdot 13^{2} + \left(3 a^{5} + 8 a^{4} + 11 a^{3} + a^{2} + 12\right)\cdot 13^{3} + \left(4 a^{5} + 7 a^{4} + 3 a^{3} + 12 a^{2} + 6 a + 12\right)\cdot 13^{4} + \left(7 a^{5} + 7 a^{3} + 8 a^{2} + 9 a + 11\right)\cdot 13^{5} + \left(7 a^{5} + 12 a^{4} + 2 a^{3}\right)\cdot 13^{6} + \left(11 a^{5} + 12 a^{4} + 12 a^{3} + 5 a^{2} + 6 a + 2\right)\cdot 13^{7} + \left(3 a^{5} + 10 a^{4} + 4 a^{3} + 12 a^{2} + 7 a + 6\right)\cdot 13^{8} +O(13^{9})$$ 3*a^5 + 3*a^4 + 10*a^3 + 5*a^2 + 11*a + 1 + (8*a^5 + a^4 + 2*a^3 + 7*a^2 + 7*a + 8)*13 + (7*a^5 + 12*a^4 + 5*a^3 + 2*a^2 + 12*a + 9)*13^2 + (3*a^5 + 8*a^4 + 11*a^3 + a^2 + 12)*13^3 + (4*a^5 + 7*a^4 + 3*a^3 + 12*a^2 + 6*a + 12)*13^4 + (7*a^5 + 7*a^3 + 8*a^2 + 9*a + 11)*13^5 + (7*a^5 + 12*a^4 + 2*a^3)*13^6 + (11*a^5 + 12*a^4 + 12*a^3 + 5*a^2 + 6*a + 2)*13^7 + (3*a^5 + 10*a^4 + 4*a^3 + 12*a^2 + 7*a + 6)*13^8+O(13^9) $r_{ 2 }$ $=$ $$9 a^{5} + 4 a^{2} + 6 a + 10 + \left(4 a^{5} + 11 a^{2} + 11 a + 4\right)\cdot 13 + \left(7 a^{5} + 11 a^{4} + 8 a^{3} + 9 a^{2} + 8 a + 4\right)\cdot 13^{2} + \left(4 a^{5} + 12 a^{4} + a^{3} + 2 a^{2} + 5 a + 12\right)\cdot 13^{3} + \left(4 a^{5} + 8 a^{4} + 5 a^{3} + 4 a^{2} + 10 a + 9\right)\cdot 13^{4} + \left(11 a^{5} + 5 a^{4} + a^{3} + 2 a^{2} + 3 a + 6\right)\cdot 13^{5} + \left(12 a^{5} + 11 a^{4} + 12 a^{3} + 12 a^{2} + 3 a + 10\right)\cdot 13^{6} + \left(2 a^{5} + 7 a^{4} + 3 a^{3} + 11 a^{2} + 9 a + 11\right)\cdot 13^{7} + \left(a^{5} + 8 a^{4} + 6 a^{3} + 2 a^{2} + 11 a + 7\right)\cdot 13^{8} +O(13^{9})$$ 9*a^5 + 4*a^2 + 6*a + 10 + (4*a^5 + 11*a^2 + 11*a + 4)*13 + (7*a^5 + 11*a^4 + 8*a^3 + 9*a^2 + 8*a + 4)*13^2 + (4*a^5 + 12*a^4 + a^3 + 2*a^2 + 5*a + 12)*13^3 + (4*a^5 + 8*a^4 + 5*a^3 + 4*a^2 + 10*a + 9)*13^4 + (11*a^5 + 5*a^4 + a^3 + 2*a^2 + 3*a + 6)*13^5 + (12*a^5 + 11*a^4 + 12*a^3 + 12*a^2 + 3*a + 10)*13^6 + (2*a^5 + 7*a^4 + 3*a^3 + 11*a^2 + 9*a + 11)*13^7 + (a^5 + 8*a^4 + 6*a^3 + 2*a^2 + 11*a + 7)*13^8+O(13^9) $r_{ 3 }$ $=$ $$5 a^{4} + 4 a^{3} + 4 a^{2} + 8 a + 2 + \left(8 a^{5} + 3 a^{4} + 11 a^{3} + 3 a^{2} + 10 a\right)\cdot 13 + \left(a^{5} + 12 a^{4} + a^{3} + a^{2} + 4 a + 1\right)\cdot 13^{2} + \left(5 a^{5} + 5 a^{4} + a^{3} + 10 a^{2} + 7\right)\cdot 13^{3} + \left(12 a^{5} + 2 a^{4} + 4 a^{3} + 2 a^{2} + 12 a\right)\cdot 13^{4} + \left(10 a^{5} + 7 a^{4} + 2 a^{3} + 11 a^{2} + 7 a + 5\right)\cdot 13^{5} + \left(8 a^{5} + a^{4} + 11 a^{3} + 10 a + 12\right)\cdot 13^{6} + \left(12 a^{5} + a^{4} + 2 a^{3} + 12 a^{2} + 2 a + 12\right)\cdot 13^{7} + \left(12 a^{5} + 11 a^{4} + 12 a^{3} + 6 a^{2} + 10 a + 3\right)\cdot 13^{8} +O(13^{9})$$ 5*a^4 + 4*a^3 + 4*a^2 + 8*a + 2 + (8*a^5 + 3*a^4 + 11*a^3 + 3*a^2 + 10*a)*13 + (a^5 + 12*a^4 + a^3 + a^2 + 4*a + 1)*13^2 + (5*a^5 + 5*a^4 + a^3 + 10*a^2 + 7)*13^3 + (12*a^5 + 2*a^4 + 4*a^3 + 2*a^2 + 12*a)*13^4 + (10*a^5 + 7*a^4 + 2*a^3 + 11*a^2 + 7*a + 5)*13^5 + (8*a^5 + a^4 + 11*a^3 + 10*a + 12)*13^6 + (12*a^5 + a^4 + 2*a^3 + 12*a^2 + 2*a + 12)*13^7 + (12*a^5 + 11*a^4 + 12*a^3 + 6*a^2 + 10*a + 3)*13^8+O(13^9) $r_{ 4 }$ $=$ $$7 a^{5} + 9 a^{4} + 7 a^{3} + 10 a^{2} + 11 a + \left(2 a^{5} + 3 a^{4} + 8 a^{3} + 2 a^{2} + 5 a + 10\right)\cdot 13 + \left(7 a^{5} + 11 a^{4} + 4 a^{3} + 2 a^{2} + 3 a + 4\right)\cdot 13^{2} + \left(a^{5} + 6 a^{4} + 2 a^{3} + 9 a^{2} + 11 a + 2\right)\cdot 13^{3} + \left(12 a^{5} + 8 a^{4} + 6 a^{3} + a^{2} + 4 a + 3\right)\cdot 13^{4} + \left(5 a^{5} + 4 a^{4} + 11 a^{3} + 9 a^{2} + 10 a + 12\right)\cdot 13^{5} + \left(9 a^{5} + 10 a^{4} + a^{3} + 12 a^{2} + 5 a + 7\right)\cdot 13^{6} + \left(2 a^{4} + 2 a^{3} + 6 a^{2} + 9 a + 9\right)\cdot 13^{7} + \left(12 a^{5} + 10 a^{3} + 5 a^{2} + 2 a + 11\right)\cdot 13^{8} +O(13^{9})$$ 7*a^5 + 9*a^4 + 7*a^3 + 10*a^2 + 11*a + (2*a^5 + 3*a^4 + 8*a^3 + 2*a^2 + 5*a + 10)*13 + (7*a^5 + 11*a^4 + 4*a^3 + 2*a^2 + 3*a + 4)*13^2 + (a^5 + 6*a^4 + 2*a^3 + 9*a^2 + 11*a + 2)*13^3 + (12*a^5 + 8*a^4 + 6*a^3 + a^2 + 4*a + 3)*13^4 + (5*a^5 + 4*a^4 + 11*a^3 + 9*a^2 + 10*a + 12)*13^5 + (9*a^5 + 10*a^4 + a^3 + 12*a^2 + 5*a + 7)*13^6 + (2*a^4 + 2*a^3 + 6*a^2 + 9*a + 9)*13^7 + (12*a^5 + 10*a^3 + 5*a^2 + 2*a + 11)*13^8+O(13^9) $r_{ 5 }$ $=$ $$6 a^{5} + 4 a^{4} + 9 a^{3} + 8 a^{2} + 9 a + 1 + \left(5 a^{5} + 2 a^{4} + 4 a^{3} + 8 a^{2} + 3\right)\cdot 13 + \left(a^{5} + 5 a^{4} + 5 a^{3} + 10 a^{2} + 5 a + 10\right)\cdot 13^{2} + \left(7 a^{5} + 9 a^{4} + 7 a^{3} + 11 a^{2} + 1\right)\cdot 13^{3} + \left(11 a^{5} + 2 a^{4} + 7 a^{3} + 3 a^{2} + 10 a + 12\right)\cdot 13^{4} + \left(7 a^{5} + 8 a^{4} + 10 a^{3} + 3 a + 1\right)\cdot 13^{5} + \left(5 a^{5} + 7 a^{4} + 8 a^{3} + 5 a^{2} + 9 a + 11\right)\cdot 13^{6} + \left(4 a^{5} + 9 a^{4} + 7 a^{3} + 9 a + 1\right)\cdot 13^{7} + \left(2 a^{5} + 7 a^{4} + 2 a^{3} + 6 a^{2} + 4 a + 7\right)\cdot 13^{8} +O(13^{9})$$ 6*a^5 + 4*a^4 + 9*a^3 + 8*a^2 + 9*a + 1 + (5*a^5 + 2*a^4 + 4*a^3 + 8*a^2 + 3)*13 + (a^5 + 5*a^4 + 5*a^3 + 10*a^2 + 5*a + 10)*13^2 + (7*a^5 + 9*a^4 + 7*a^3 + 11*a^2 + 1)*13^3 + (11*a^5 + 2*a^4 + 7*a^3 + 3*a^2 + 10*a + 12)*13^4 + (7*a^5 + 8*a^4 + 10*a^3 + 3*a + 1)*13^5 + (5*a^5 + 7*a^4 + 8*a^3 + 5*a^2 + 9*a + 11)*13^6 + (4*a^5 + 9*a^4 + 7*a^3 + 9*a + 1)*13^7 + (2*a^5 + 7*a^4 + 2*a^3 + 6*a^2 + 4*a + 7)*13^8+O(13^9) $r_{ 6 }$ $=$ $$11 a^{5} + 6 a^{4} + 3 a^{3} + 8 a^{2} + 4 a + 5 + \left(12 a^{5} + 2 a^{4} + 6 a^{3} + 12 a^{2} + 11 a + 7\right)\cdot 13 + \left(a^{5} + 10 a^{3} + 6 a^{2} + 4 a + 9\right)\cdot 13^{2} + \left(10 a^{5} + 10 a^{4} + 2 a^{3} + 12 a^{2} + 9\right)\cdot 13^{3} + \left(5 a^{4} + 8 a^{3} + 5 a^{2} + 4 a + 3\right)\cdot 13^{4} + \left(12 a^{4} + a^{3} + 2 a + 8\right)\cdot 13^{5} + \left(3 a^{5} + 3 a^{4} + 2 a^{3} + 5 a^{2} + 8 a + 1\right)\cdot 13^{6} + \left(4 a^{5} + 8 a^{4} + 7 a^{3} + 6 a^{2} + a + 12\right)\cdot 13^{7} + \left(6 a^{5} + 6 a^{4} + 4 a^{3} + 3 a^{2} + 9 a + 11\right)\cdot 13^{8} +O(13^{9})$$ 11*a^5 + 6*a^4 + 3*a^3 + 8*a^2 + 4*a + 5 + (12*a^5 + 2*a^4 + 6*a^3 + 12*a^2 + 11*a + 7)*13 + (a^5 + 10*a^3 + 6*a^2 + 4*a + 9)*13^2 + (10*a^5 + 10*a^4 + 2*a^3 + 12*a^2 + 9)*13^3 + (5*a^4 + 8*a^3 + 5*a^2 + 4*a + 3)*13^4 + (12*a^4 + a^3 + 2*a + 8)*13^5 + (3*a^5 + 3*a^4 + 2*a^3 + 5*a^2 + 8*a + 1)*13^6 + (4*a^5 + 8*a^4 + 7*a^3 + 6*a^2 + a + 12)*13^7 + (6*a^5 + 6*a^4 + 4*a^3 + 3*a^2 + 9*a + 11)*13^8+O(13^9) $r_{ 7 }$ $=$ $$12 a^{5} + 12 a^{4} + 11 a^{3} + a^{2} + a + 5 + \left(8 a^{5} + 4 a^{4} + 9 a^{3} + 8 a^{2} + 6 a + 5\right)\cdot 13 + \left(2 a^{5} + 10 a^{4} + 5 a^{3} + 5 a^{2} + 3 a + 6\right)\cdot 13^{2} + \left(12 a^{5} + a^{3} + 2 a + 10\right)\cdot 13^{3} + \left(10 a^{5} + 12 a^{4} + 9 a^{3} + a^{2} + 2 a + 8\right)\cdot 13^{4} + \left(12 a^{5} + 5 a^{4} + 9 a^{3} + a^{2} + 6 a + 9\right)\cdot 13^{5} + \left(8 a^{5} + 9 a^{4} + 10 a^{3} + 8 a^{2} + 8 a + 7\right)\cdot 13^{6} + \left(6 a^{5} + 2 a^{4} + 3 a^{3} + 5 a^{2} + 4 a + 9\right)\cdot 13^{7} + \left(8 a^{4} + 2 a^{3} + 5 a^{2} + 11 a + 8\right)\cdot 13^{8} +O(13^{9})$$ 12*a^5 + 12*a^4 + 11*a^3 + a^2 + a + 5 + (8*a^5 + 4*a^4 + 9*a^3 + 8*a^2 + 6*a + 5)*13 + (2*a^5 + 10*a^4 + 5*a^3 + 5*a^2 + 3*a + 6)*13^2 + (12*a^5 + a^3 + 2*a + 10)*13^3 + (10*a^5 + 12*a^4 + 9*a^3 + a^2 + 2*a + 8)*13^4 + (12*a^5 + 5*a^4 + 9*a^3 + a^2 + 6*a + 9)*13^5 + (8*a^5 + 9*a^4 + 10*a^3 + 8*a^2 + 8*a + 7)*13^6 + (6*a^5 + 2*a^4 + 3*a^3 + 5*a^2 + 4*a + 9)*13^7 + (8*a^4 + 2*a^3 + 5*a^2 + 11*a + 8)*13^8+O(13^9) $r_{ 8 }$ $=$ $$12 a^{5} + 2 a^{4} + 5 a^{3} + a^{2} + 3 a + 10 + \left(10 a^{5} + 2 a^{4} + 9 a^{2} + 3\right)\cdot 13 + \left(2 a^{5} + 7 a^{4} + 7 a^{3} + 5 a + 12\right)\cdot 13^{2} + \left(4 a^{5} + 6 a^{4} + 2 a^{3} + 3 a^{2} + 11 a + 10\right)\cdot 13^{3} + \left(9 a^{5} + 3 a^{4} + 6 a^{3} + 5 a^{2} + 5 a + 9\right)\cdot 13^{4} + \left(3 a^{5} + 6 a^{4} + 3 a^{3} + 9 a^{2} + 5\right)\cdot 13^{5} + \left(2 a^{5} + 7 a^{4} + 7 a^{3} + 6 a + 5\right)\cdot 13^{6} + \left(9 a^{5} + 8 a^{4} + 2 a^{3} + 10 a + 10\right)\cdot 13^{7} + \left(9 a^{5} + 4 a^{4} + 9 a^{3} + 5 a^{2} + 1\right)\cdot 13^{8} +O(13^{9})$$ 12*a^5 + 2*a^4 + 5*a^3 + a^2 + 3*a + 10 + (10*a^5 + 2*a^4 + 9*a^2 + 3)*13 + (2*a^5 + 7*a^4 + 7*a^3 + 5*a + 12)*13^2 + (4*a^5 + 6*a^4 + 2*a^3 + 3*a^2 + 11*a + 10)*13^3 + (9*a^5 + 3*a^4 + 6*a^3 + 5*a^2 + 5*a + 9)*13^4 + (3*a^5 + 6*a^4 + 3*a^3 + 9*a^2 + 5)*13^5 + (2*a^5 + 7*a^4 + 7*a^3 + 6*a + 5)*13^6 + (9*a^5 + 8*a^4 + 2*a^3 + 10*a + 10)*13^7 + (9*a^5 + 4*a^4 + 9*a^3 + 5*a^2 + 1)*13^8+O(13^9) $r_{ 9 }$ $=$ $$6 a^{5} + 3 a^{4} + 3 a^{3} + 3 a^{2} + 2 a + 7 + \left(4 a^{5} + 4 a^{4} + 4 a^{3} + 3 a^{2} + 9 a + 3\right)\cdot 13 + \left(9 a^{5} + 11 a^{4} + 4 a^{3} + 9 a^{2} + 10 a + 12\right)\cdot 13^{2} + \left(11 a^{5} + 9 a^{4} + a^{3} + 12 a^{2} + 3 a + 1\right)\cdot 13^{3} + \left(12 a^{5} + 11 a^{3} + 7 a + 6\right)\cdot 13^{4} + \left(9 a^{5} + 7 a^{4} + 2 a^{3} + 11 a^{2} + 10 a + 4\right)\cdot 13^{5} + \left(8 a^{5} + 9 a^{4} + 2 a^{3} + 12 a^{2} + 7 a + 10\right)\cdot 13^{6} + \left(a^{5} + 10 a^{4} + 12 a^{3} + 6 a^{2} + 11 a + 11\right)\cdot 13^{7} + \left(11 a^{5} + 3 a^{4} + 7 a^{3} + 2 a^{2} + 3 a + 9\right)\cdot 13^{8} +O(13^{9})$$ 6*a^5 + 3*a^4 + 3*a^3 + 3*a^2 + 2*a + 7 + (4*a^5 + 4*a^4 + 4*a^3 + 3*a^2 + 9*a + 3)*13 + (9*a^5 + 11*a^4 + 4*a^3 + 9*a^2 + 10*a + 12)*13^2 + (11*a^5 + 9*a^4 + a^3 + 12*a^2 + 3*a + 1)*13^3 + (12*a^5 + 11*a^3 + 7*a + 6)*13^4 + (9*a^5 + 7*a^4 + 2*a^3 + 11*a^2 + 10*a + 4)*13^5 + (8*a^5 + 9*a^4 + 2*a^3 + 12*a^2 + 7*a + 10)*13^6 + (a^5 + 10*a^4 + 12*a^3 + 6*a^2 + 11*a + 11)*13^7 + (11*a^5 + 3*a^4 + 7*a^3 + 2*a^2 + 3*a + 9)*13^8+O(13^9) $r_{ 10 }$ $=$ $$a^{5} + 12 a^{4} + 12 a^{3} + a^{2} + 11 a + 5 + \left(12 a^{5} + 12 a^{4} + 6 a^{3} + 8 a^{2} + 11 a + 5\right)\cdot 13 + \left(10 a^{5} + 12 a^{4} + 7 a^{3} + 4 a^{2} + 11 a + 7\right)\cdot 13^{2} + \left(12 a^{5} + 12 a^{4} + 11 a^{3} + a^{2} + 3 a + 10\right)\cdot 13^{3} + \left(11 a^{5} + 6 a^{4} + 4 a^{3} + 8 a^{2} + 3\right)\cdot 13^{4} + \left(8 a^{5} + 12 a^{4} + 10 a^{2} + 2 a + 7\right)\cdot 13^{5} + \left(12 a^{5} + 4 a^{4} + 7 a^{3} + 7 a^{2} + 12 a + 8\right)\cdot 13^{6} + \left(7 a^{5} + 9 a^{4} + 2 a^{2} + 10 a + 3\right)\cdot 13^{7} + \left(2 a^{5} + 9 a^{4} + 11 a^{3} + a^{2} + 4 a + 12\right)\cdot 13^{8} +O(13^{9})$$ a^5 + 12*a^4 + 12*a^3 + a^2 + 11*a + 5 + (12*a^5 + 12*a^4 + 6*a^3 + 8*a^2 + 11*a + 5)*13 + (10*a^5 + 12*a^4 + 7*a^3 + 4*a^2 + 11*a + 7)*13^2 + (12*a^5 + 12*a^4 + 11*a^3 + a^2 + 3*a + 10)*13^3 + (11*a^5 + 6*a^4 + 4*a^3 + 8*a^2 + 3)*13^4 + (8*a^5 + 12*a^4 + 10*a^2 + 2*a + 7)*13^5 + (12*a^5 + 4*a^4 + 7*a^3 + 7*a^2 + 12*a + 8)*13^6 + (7*a^5 + 9*a^4 + 2*a^2 + 10*a + 3)*13^7 + (2*a^5 + 9*a^4 + 11*a^3 + a^2 + 4*a + 12)*13^8+O(13^9) $r_{ 11 }$ $=$ $$6 a^{5} + 6 a^{4} + 4 a^{3} + 11 a + 8 + \left(6 a^{5} + 12 a^{4} + 3 a^{3} + 12\right)\cdot 13 + \left(8 a^{5} + 11 a^{4} + 2 a^{3} + 11 a^{2} + 3 a\right)\cdot 13^{2} + \left(6 a^{4} + 2 a^{3} + 6 a^{2} + 6 a\right)\cdot 13^{3} + \left(4 a^{5} + 4 a^{4} + 5 a^{3} + 8 a^{2} + 4 a + 1\right)\cdot 13^{4} + \left(8 a^{5} + 12 a^{4} + 11 a^{3} + 9 a^{2} + 4 a + 12\right)\cdot 13^{5} + \left(6 a^{5} + 4 a^{4} + 7 a^{3} + 12 a^{2} + 6 a + 4\right)\cdot 13^{6} + \left(11 a^{5} + 5 a^{4} + 9 a^{2} + 7 a + 11\right)\cdot 13^{7} + \left(10 a^{5} + 6 a^{4} + 8 a^{3} + 3 a^{2} + 12 a + 9\right)\cdot 13^{8} +O(13^{9})$$ 6*a^5 + 6*a^4 + 4*a^3 + 11*a + 8 + (6*a^5 + 12*a^4 + 3*a^3 + 12)*13 + (8*a^5 + 11*a^4 + 2*a^3 + 11*a^2 + 3*a)*13^2 + (6*a^4 + 2*a^3 + 6*a^2 + 6*a)*13^3 + (4*a^5 + 4*a^4 + 5*a^3 + 8*a^2 + 4*a + 1)*13^4 + (8*a^5 + 12*a^4 + 11*a^3 + 9*a^2 + 4*a + 12)*13^5 + (6*a^5 + 4*a^4 + 7*a^3 + 12*a^2 + 6*a + 4)*13^6 + (11*a^5 + 5*a^4 + 9*a^2 + 7*a + 11)*13^7 + (10*a^5 + 6*a^4 + 8*a^3 + 3*a^2 + 12*a + 9)*13^8+O(13^9) $r_{ 12 }$ $=$ $$5 a^{5} + 3 a^{4} + 10 a^{3} + 7 a^{2} + a + 2 + \left(6 a^{5} + 2 a^{4} + 6 a^{3} + 3 a^{2} + 2 a + 1\right)\cdot 13 + \left(3 a^{5} + 11 a^{4} + 2 a^{3} + 4 a + 12\right)\cdot 13^{2} + \left(4 a^{5} + 12 a^{4} + 6 a^{3} + 6 a^{2} + 5 a + 10\right)\cdot 13^{3} + \left(9 a^{5} + 6 a^{3} + 10 a^{2} + 10 a + 5\right)\cdot 13^{4} + \left(3 a^{5} + 8 a^{4} + 2 a^{3} + 3 a^{2} + 3 a + 5\right)\cdot 13^{5} + \left(4 a^{5} + 7 a^{4} + 4 a^{3} + 12 a^{2} + 12 a + 9\right)\cdot 13^{6} + \left(4 a^{5} + 11 a^{4} + 9 a^{3} + 9 a^{2} + 6 a + 6\right)\cdot 13^{7} + \left(4 a^{5} + 12 a^{4} + 11 a^{3} + 9 a^{2} + 11 a + 12\right)\cdot 13^{8} +O(13^{9})$$ 5*a^5 + 3*a^4 + 10*a^3 + 7*a^2 + a + 2 + (6*a^5 + 2*a^4 + 6*a^3 + 3*a^2 + 2*a + 1)*13 + (3*a^5 + 11*a^4 + 2*a^3 + 4*a + 12)*13^2 + (4*a^5 + 12*a^4 + 6*a^3 + 6*a^2 + 5*a + 10)*13^3 + (9*a^5 + 6*a^3 + 10*a^2 + 10*a + 5)*13^4 + (3*a^5 + 8*a^4 + 2*a^3 + 3*a^2 + 3*a + 5)*13^5 + (4*a^5 + 7*a^4 + 4*a^3 + 12*a^2 + 12*a + 9)*13^6 + (4*a^5 + 11*a^4 + 9*a^3 + 9*a^2 + 6*a + 6)*13^7 + (4*a^5 + 12*a^4 + 11*a^3 + 9*a^2 + 11*a + 12)*13^8+O(13^9)

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

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

### Character values on conjugacy classes

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