# Properties

 Label 1.72.6t1.d Dimension $1$ Group $C_6$ Conductor $72$ Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $$72$$$$\medspace = 2^{3} \cdot 3^{2}$$ Artin number field: Galois closure of 6.6.3359232.1 Galois orbit size: $2$ Smallest permutation container: $C_6$ Parity: even Projective image: $C_1$ Projective field: $$\Q$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $$x^{2} + 18 x + 2$$
Roots:
 $r_{ 1 }$ $=$ $$14 a + 12 + \left(11 a + 10\right)\cdot 19 + \left(11 a + 9\right)\cdot 19^{2} + \left(12 a + 18\right)\cdot 19^{3} + \left(14 a + 17\right)\cdot 19^{4} +O(19^{5})$$ $r_{ 2 }$ $=$ $$18 a + 10 + 8\cdot 19 + \left(5 a + 7\right)\cdot 19^{2} + \left(a + 11\right)\cdot 19^{3} + \left(16 a + 11\right)\cdot 19^{4} +O(19^{5})$$ $r_{ 3 }$ $=$ $$15 a + 2 + \left(10 a + 2\right)\cdot 19 + \left(6 a + 2\right)\cdot 19^{2} + \left(11 a + 7\right)\cdot 19^{3} + \left(17 a + 6\right)\cdot 19^{4} +O(19^{5})$$ $r_{ 4 }$ $=$ $$5 a + 7 + \left(7 a + 8\right)\cdot 19 + \left(7 a + 9\right)\cdot 19^{2} + 6 a\cdot 19^{3} + \left(4 a + 1\right)\cdot 19^{4} +O(19^{5})$$ $r_{ 5 }$ $=$ $$a + 9 + \left(18 a + 10\right)\cdot 19 + \left(13 a + 11\right)\cdot 19^{2} + \left(17 a + 7\right)\cdot 19^{3} + \left(2 a + 7\right)\cdot 19^{4} +O(19^{5})$$ $r_{ 6 }$ $=$ $$4 a + 17 + \left(8 a + 16\right)\cdot 19 + \left(12 a + 16\right)\cdot 19^{2} + \left(7 a + 11\right)\cdot 19^{3} + \left(a + 12\right)\cdot 19^{4} +O(19^{5})$$

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

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

### Character values on conjugacy classes

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