# Properties

 Label 2.273.6t5.d Dimension $2$ Group $S_3\times C_3$ Conductor $273$ Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3\times C_3$ Conductor: $$273$$$$\medspace = 3 \cdot 7 \cdot 13$$ Artin number field: Galois closure of 6.0.223587.2 Galois orbit size: $2$ Smallest permutation container: $S_3\times C_3$ Parity: odd Projective image: $S_3$ Projective field: 3.1.24843.1

## Galois action

### Roots of defining polynomial

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

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

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

### Character values on conjugacy classes

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