# Properties

 Label 1.155.6t1.b Dimension $1$ Group $C_6$ Conductor $155$ Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $$155$$$$\medspace = 5 \cdot 31$$ Artin number field: Galois closure of 6.0.3578643875.1 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 $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $$x^{2} + 24x + 2$$
Roots:
 $r_{ 1 }$ $=$ $$5 a + 11 + \left(14 a + 13\right)\cdot 29 + \left(26 a + 16\right)\cdot 29^{2} + \left(15 a + 19\right)\cdot 29^{3} + \left(19 a + 5\right)\cdot 29^{4} +O(29^{5})$$ 5*a + 11 + (14*a + 13)*29 + (26*a + 16)*29^2 + (15*a + 19)*29^3 + (19*a + 5)*29^4+O(29^5) $r_{ 2 }$ $=$ $$7 a + 14 + \left(a + 10\right)\cdot 29 + \left(26 a + 10\right)\cdot 29^{2} + \left(9 a + 27\right)\cdot 29^{3} + \left(28 a + 2\right)\cdot 29^{4} +O(29^{5})$$ 7*a + 14 + (a + 10)*29 + (26*a + 10)*29^2 + (9*a + 27)*29^3 + (28*a + 2)*29^4+O(29^5) $r_{ 3 }$ $=$ $$26 a + 11 + 12\cdot 29 + \left(15 a + 15\right)\cdot 29^{2} + \left(a + 5\right)\cdot 29^{3} + \left(8 a + 10\right)\cdot 29^{4} +O(29^{5})$$ 26*a + 11 + 12*29 + (15*a + 15)*29^2 + (a + 5)*29^3 + (8*a + 10)*29^4+O(29^5) $r_{ 4 }$ $=$ $$3 a + 25 + \left(28 a + 19\right)\cdot 29 + \left(13 a + 2\right)\cdot 29^{2} + \left(27 a + 27\right)\cdot 29^{3} + \left(20 a + 19\right)\cdot 29^{4} +O(29^{5})$$ 3*a + 25 + (28*a + 19)*29 + (13*a + 2)*29^2 + (27*a + 27)*29^3 + (20*a + 19)*29^4+O(29^5) $r_{ 5 }$ $=$ $$24 a + 7 + \left(14 a + 21\right)\cdot 29 + \left(2 a + 18\right)\cdot 29^{2} + \left(13 a + 14\right)\cdot 29^{3} + 9 a\cdot 29^{4} +O(29^{5})$$ 24*a + 7 + (14*a + 21)*29 + (2*a + 18)*29^2 + (13*a + 14)*29^3 + 9*a*29^4+O(29^5) $r_{ 6 }$ $=$ $$22 a + 20 + \left(27 a + 9\right)\cdot 29 + \left(2 a + 23\right)\cdot 29^{2} + \left(19 a + 21\right)\cdot 29^{3} + 18\cdot 29^{4} +O(29^{5})$$ 22*a + 20 + (27*a + 9)*29 + (2*a + 23)*29^2 + (19*a + 21)*29^3 + 18*29^4+O(29^5)

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

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

### Character values on conjugacy classes

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