# Properties

 Label 1.155.6t1.b.a Dimension $1$ Group $C_6$ Conductor $155$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $$155$$$$\medspace = 5 \cdot 31$$ Artin field: 6.0.3578643875.1 Galois orbit size: $2$ Smallest permutation container: $C_6$ Parity: odd Dirichlet character: $$\chi_{155}(99,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - x^{5} + 34 x^{4} - 73 x^{3} + 509 x^{2} - 470 x + 1396$$  .

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $$x^{2} + 24 x + 2$$

Roots:
 $r_{ 1 }$ $=$ $$5 a + 11 + \left(14 a + 13\right)\cdot 29 + \left(26 a + 16\right)\cdot 29^{2} + \left(15 a + 19\right)\cdot 29^{3} + \left(19 a + 5\right)\cdot 29^{4} +O(29^{5})$$ $r_{ 2 }$ $=$ $$7 a + 14 + \left(a + 10\right)\cdot 29 + \left(26 a + 10\right)\cdot 29^{2} + \left(9 a + 27\right)\cdot 29^{3} + \left(28 a + 2\right)\cdot 29^{4} +O(29^{5})$$ $r_{ 3 }$ $=$ $$26 a + 11 + 12\cdot 29 + \left(15 a + 15\right)\cdot 29^{2} + \left(a + 5\right)\cdot 29^{3} + \left(8 a + 10\right)\cdot 29^{4} +O(29^{5})$$ $r_{ 4 }$ $=$ $$3 a + 25 + \left(28 a + 19\right)\cdot 29 + \left(13 a + 2\right)\cdot 29^{2} + \left(27 a + 27\right)\cdot 29^{3} + \left(20 a + 19\right)\cdot 29^{4} +O(29^{5})$$ $r_{ 5 }$ $=$ $$24 a + 7 + \left(14 a + 21\right)\cdot 29 + \left(2 a + 18\right)\cdot 29^{2} + \left(13 a + 14\right)\cdot 29^{3} + 9 a\cdot 29^{4} +O(29^{5})$$ $r_{ 6 }$ $=$ $$22 a + 20 + \left(27 a + 9\right)\cdot 29 + \left(2 a + 23\right)\cdot 29^{2} + \left(19 a + 21\right)\cdot 29^{3} + 18\cdot 29^{4} +O(29^{5})$$

## 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 value $1$ $1$ $()$ $1$ $1$ $2$ $(1,5)(2,6)(3,4)$ $-1$ $1$ $3$ $(1,3,6)(2,5,4)$ $\zeta_{3}$ $1$ $3$ $(1,6,3)(2,4,5)$ $-\zeta_{3} - 1$ $1$ $6$ $(1,2,3,5,6,4)$ $\zeta_{3} + 1$ $1$ $6$ $(1,4,6,5,3,2)$ $-\zeta_{3}$

The blue line marks the conjugacy class containing complex conjugation.