# Properties

 Label 1.65.6t1.b Dimension $1$ Group $C_6$ Conductor $65$ Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $$65$$$$\medspace = 5 \cdot 13$$ Artin number field: Galois closure of 6.6.3570125.1 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 $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $$x^{2} + 45 x + 5$$
Roots:
 $r_{ 1 }$ $=$ $$19 a + 27 + \left(21 a + 35\right)\cdot 47 + \left(29 a + 24\right)\cdot 47^{2} + \left(9 a + 7\right)\cdot 47^{3} + \left(30 a + 17\right)\cdot 47^{4} +O(47^{5})$$ $r_{ 2 }$ $=$ $$19 a + 43 + \left(21 a + 23\right)\cdot 47 + \left(29 a + 24\right)\cdot 47^{2} + \left(9 a + 34\right)\cdot 47^{3} + \left(30 a + 37\right)\cdot 47^{4} +O(47^{5})$$ $r_{ 3 }$ $=$ $$28 a + 34 + 25 a\cdot 47 + \left(17 a + 15\right)\cdot 47^{2} + \left(37 a + 24\right)\cdot 47^{3} + \left(16 a + 41\right)\cdot 47^{4} +O(47^{5})$$ $r_{ 4 }$ $=$ $$28 a + 29 + \left(25 a + 22\right)\cdot 47 + \left(17 a + 2\right)\cdot 47^{2} + \left(37 a + 10\right)\cdot 47^{3} + \left(16 a + 37\right)\cdot 47^{4} +O(47^{5})$$ $r_{ 5 }$ $=$ $$28 a + 18 + \left(25 a + 12\right)\cdot 47 + \left(17 a + 15\right)\cdot 47^{2} + \left(37 a + 44\right)\cdot 47^{3} + \left(16 a + 20\right)\cdot 47^{4} +O(47^{5})$$ $r_{ 6 }$ $=$ $$19 a + 38 + \left(21 a + 45\right)\cdot 47 + \left(29 a + 11\right)\cdot 47^{2} + \left(9 a + 20\right)\cdot 47^{3} + \left(30 a + 33\right)\cdot 47^{4} +O(47^{5})$$

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

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

### Character values on conjugacy classes

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