Properties

 Label 2.3040.12t18.c.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 12.0.34162868224000000.1 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

 $f(x)$ $=$ $$x^{12} - 4 x^{11} + 10 x^{10} - 16 x^{9} + 28 x^{8} - 36 x^{7} + 58 x^{6} - 52 x^{5} + 79 x^{4} + \cdots + 41$$ x^12 - 4*x^11 + 10*x^10 - 16*x^9 + 28*x^8 - 36*x^7 + 58*x^6 - 52*x^5 + 79*x^4 - 60*x^3 + 92*x^2 - 52*x + 41 .

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 8.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $$x^{6} + 10x^{3} + 11x^{2} + 11x + 2$$

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.