# Properties

 Label 2.180.6t5.b Dimension $2$ Group $S_3\times C_3$ Conductor $180$ Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3\times C_3$ Conductor: $$180$$$$\medspace = 2^{2} \cdot 3^{2} \cdot 5$$ Artin number field: Galois closure of 6.0.648000.1 Galois orbit size: $2$ Smallest permutation container: $S_3\times C_3$ Parity: odd Projective image: $S_3$ Projective field: Galois closure of 3.1.1620.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $$x^{2} + 16x + 3$$
Roots:
 $r_{ 1 }$ $=$ $$3 a + 10 + \left(16 a + 2\right)\cdot 17 + \left(5 a + 6\right)\cdot 17^{2} + \left(4 a + 12\right)\cdot 17^{3} + \left(6 a + 16\right)\cdot 17^{4} +O(17^{5})$$ 3*a + 10 + (16*a + 2)*17 + (5*a + 6)*17^2 + (4*a + 12)*17^3 + (6*a + 16)*17^4+O(17^5) $r_{ 2 }$ $=$ $$5 a + 2 + \left(16 a + 1\right)\cdot 17 + \left(10 a + 8\right)\cdot 17^{2} + \left(9 a + 13\right)\cdot 17^{3} + \left(a + 5\right)\cdot 17^{4} +O(17^{5})$$ 5*a + 2 + (16*a + 1)*17 + (10*a + 8)*17^2 + (9*a + 13)*17^3 + (a + 5)*17^4+O(17^5) $r_{ 3 }$ $=$ $$12 a + 7 + 12\cdot 17 + \left(6 a + 2\right)\cdot 17^{2} + \left(7 a + 12\right)\cdot 17^{3} + \left(15 a + 14\right)\cdot 17^{4} +O(17^{5})$$ 12*a + 7 + 12*17 + (6*a + 2)*17^2 + (7*a + 12)*17^3 + (15*a + 14)*17^4+O(17^5) $r_{ 4 }$ $=$ $$14 a + 13 + 15\cdot 17 + \left(11 a + 12\right)\cdot 17^{2} + \left(12 a + 10\right)\cdot 17^{3} + \left(10 a + 1\right)\cdot 17^{4} +O(17^{5})$$ 14*a + 13 + 15*17 + (11*a + 12)*17^2 + (12*a + 10)*17^3 + (10*a + 1)*17^4+O(17^5) $r_{ 5 }$ $=$ $$3 a + 9 + 4 a\cdot 17 + \left(10 a + 16\right)\cdot 17^{2} + \left(11 a + 8\right)\cdot 17^{3} + \left(3 a + 1\right)\cdot 17^{4} +O(17^{5})$$ 3*a + 9 + 4*a*17 + (10*a + 16)*17^2 + (11*a + 8)*17^3 + (3*a + 1)*17^4+O(17^5) $r_{ 6 }$ $=$ $$14 a + 12 + \left(12 a + 1\right)\cdot 17 + \left(6 a + 5\right)\cdot 17^{2} + \left(5 a + 10\right)\cdot 17^{3} + \left(13 a + 10\right)\cdot 17^{4} +O(17^{5})$$ 14*a + 12 + (12*a + 1)*17 + (6*a + 5)*17^2 + (5*a + 10)*17^3 + (13*a + 10)*17^4+O(17^5)

### Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

 Cycle notation $(1,6,2)$ $(3,5,4)$ $(1,5)(2,4)(3,6)$

### Character values on conjugacy classes

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