# Properties

 Label 1.260.6t1.b Dimension $1$ Group $C_6$ Conductor $260$ Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $$260$$$$\medspace = 2^{2} \cdot 5 \cdot 13$$ Artin number field: Galois closure of 6.0.2970344000.1 Galois orbit size: $2$ Smallest permutation container: $C_6$ Parity: odd Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## 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 $\mathbb{Q}_{ 47 }$: $$x^{2} + 45x + 5$$
Roots:
 $r_{ 1 }$ $=$ $$14 a + 33 + \left(5 a + 1\right)\cdot 47 + \left(17 a + 9\right)\cdot 47^{2} + \left(20 a + 35\right)\cdot 47^{3} + \left(23 a + 33\right)\cdot 47^{4} +O(47^{5})$$ 14*a + 33 + (5*a + 1)*47 + (17*a + 9)*47^2 + (20*a + 35)*47^3 + (23*a + 33)*47^4+O(47^5) $r_{ 2 }$ $=$ $$8 a + 39 + \left(14 a + 36\right)\cdot 47 + \left(36 a + 17\right)\cdot 47^{2} + \left(37 a + 27\right)\cdot 47^{3} + \left(25 a + 16\right)\cdot 47^{4} +O(47^{5})$$ 8*a + 39 + (14*a + 36)*47 + (36*a + 17)*47^2 + (37*a + 27)*47^3 + (25*a + 16)*47^4+O(47^5) $r_{ 3 }$ $=$ $$12 a + 35 + \left(17 a + 35\right)\cdot 47 + \left(28 a + 3\right)\cdot 47^{2} + \left(44 a + 40\right)\cdot 47^{3} + \left(29 a + 15\right)\cdot 47^{4} +O(47^{5})$$ 12*a + 35 + (17*a + 35)*47 + (28*a + 3)*47^2 + (44*a + 40)*47^3 + (29*a + 15)*47^4+O(47^5) $r_{ 4 }$ $=$ $$33 a + 14 + \left(41 a + 45\right)\cdot 47 + \left(29 a + 37\right)\cdot 47^{2} + \left(26 a + 11\right)\cdot 47^{3} + \left(23 a + 13\right)\cdot 47^{4} +O(47^{5})$$ 33*a + 14 + (41*a + 45)*47 + (29*a + 37)*47^2 + (26*a + 11)*47^3 + (23*a + 13)*47^4+O(47^5) $r_{ 5 }$ $=$ $$39 a + 8 + \left(32 a + 10\right)\cdot 47 + \left(10 a + 29\right)\cdot 47^{2} + \left(9 a + 19\right)\cdot 47^{3} + \left(21 a + 30\right)\cdot 47^{4} +O(47^{5})$$ 39*a + 8 + (32*a + 10)*47 + (10*a + 29)*47^2 + (9*a + 19)*47^3 + (21*a + 30)*47^4+O(47^5) $r_{ 6 }$ $=$ $$35 a + 12 + \left(29 a + 11\right)\cdot 47 + \left(18 a + 43\right)\cdot 47^{2} + \left(2 a + 6\right)\cdot 47^{3} + \left(17 a + 31\right)\cdot 47^{4} +O(47^{5})$$ 35*a + 12 + (29*a + 11)*47 + (18*a + 43)*47^2 + (2*a + 6)*47^3 + (17*a + 31)*47^4+O(47^5)

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

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

### Character values on conjugacy classes

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