# Properties

 Label 1.171.6t1.f Dimension $1$ Group $C_6$ Conductor $171$ Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $$171$$$$\medspace = 3^{2} \cdot 19$$ Artin number field: Galois closure of 6.0.45001899.1 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 $\Q_{ 37 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $$x^{2} + 33x + 2$$
Roots:
 $r_{ 1 }$ $=$ $$20 a + 21 + \left(11 a + 25\right)\cdot 37 + \left(26 a + 31\right)\cdot 37^{2} + \left(3 a + 18\right)\cdot 37^{3} + \left(31 a + 3\right)\cdot 37^{4} +O(37^{5})$$ 20*a + 21 + (11*a + 25)*37 + (26*a + 31)*37^2 + (3*a + 18)*37^3 + (31*a + 3)*37^4+O(37^5) $r_{ 2 }$ $=$ $$17 a + 31 + \left(25 a + 32\right)\cdot 37 + \left(10 a + 3\right)\cdot 37^{2} + \left(33 a + 12\right)\cdot 37^{3} + \left(5 a + 1\right)\cdot 37^{4} +O(37^{5})$$ 17*a + 31 + (25*a + 32)*37 + (10*a + 3)*37^2 + (33*a + 12)*37^3 + (5*a + 1)*37^4+O(37^5) $r_{ 3 }$ $=$ $$20 a + 2 + \left(11 a + 21\right)\cdot 37 + \left(26 a + 28\right)\cdot 37^{2} + \left(3 a + 11\right)\cdot 37^{3} + \left(31 a + 27\right)\cdot 37^{4} +O(37^{5})$$ 20*a + 2 + (11*a + 21)*37 + (26*a + 28)*37^2 + (3*a + 11)*37^3 + (31*a + 27)*37^4+O(37^5) $r_{ 4 }$ $=$ $$17 a + 27 + \left(25 a + 14\right)\cdot 37 + \left(10 a + 14\right)\cdot 37^{2} + \left(33 a + 7\right)\cdot 37^{3} + \left(5 a + 13\right)\cdot 37^{4} +O(37^{5})$$ 17*a + 27 + (25*a + 14)*37 + (10*a + 14)*37^2 + (33*a + 7)*37^3 + (5*a + 13)*37^4+O(37^5) $r_{ 5 }$ $=$ $$20 a + 25 + \left(11 a + 6\right)\cdot 37 + \left(26 a + 21\right)\cdot 37^{2} + \left(3 a + 23\right)\cdot 37^{3} + \left(31 a + 28\right)\cdot 37^{4} +O(37^{5})$$ 20*a + 25 + (11*a + 6)*37 + (26*a + 21)*37^2 + (3*a + 23)*37^3 + (31*a + 28)*37^4+O(37^5) $r_{ 6 }$ $=$ $$17 a + 8 + \left(25 a + 10\right)\cdot 37 + \left(10 a + 11\right)\cdot 37^{2} + 33 a\cdot 37^{3} + 5 a\cdot 37^{4} +O(37^{5})$$ 17*a + 8 + (25*a + 10)*37 + (10*a + 11)*37^2 + 33*a*37^3 + 5*a*37^4+O(37^5)

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

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

### Character values on conjugacy classes

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