# Properties

 Label 1.111.6t1.b.a Dimension 1 Group $C_6$ Conductor $3 \cdot 37$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $111= 3 \cdot 37$ Artin number field: Splitting field of 6.0.50602347.1 defined by $f= x^{6} - x^{5} + 13 x^{4} + 34 x^{3} + 133 x^{2} + 132 x + 121$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_6$ Parity: Odd Corresponding Dirichlet character: $$\chi_{111}(47,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## 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 $\mathbb{Q}_{ 29 }$: $x^{2} + 24 x + 2$
Roots:
 $r_{ 1 }$ $=$ $16 a + 5 + 13\cdot 29 + \left(5 a + 4\right)\cdot 29^{2} + \left(27 a + 2\right)\cdot 29^{3} + \left(27 a + 27\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ $r_{ 2 }$ $=$ $18 a + 24 + \left(24 a + 24\right)\cdot 29 + 28\cdot 29^{2} + \left(19 a + 21\right)\cdot 29^{3} + \left(27 a + 26\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ $r_{ 3 }$ $=$ $13 a + 27 + \left(28 a + 28\right)\cdot 29 + \left(23 a + 28\right)\cdot 29^{2} + \left(a + 16\right)\cdot 29^{3} + \left(a + 23\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ $r_{ 4 }$ $=$ $17 a + 18 + \left(7 a + 21\right)\cdot 29 + 10 a\cdot 29^{2} + \left(10 a + 2\right)\cdot 29^{3} + 15 a\cdot 29^{4} +O\left(29^{ 5 }\right)$ $r_{ 5 }$ $=$ $11 a + 27 + \left(4 a + 13\right)\cdot 29 + \left(28 a + 8\right)\cdot 29^{2} + 9 a\cdot 29^{3} + \left(a + 1\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ $r_{ 6 }$ $=$ $12 a + 16 + \left(21 a + 13\right)\cdot 29 + \left(18 a + 15\right)\cdot 29^{2} + \left(18 a + 14\right)\cdot 29^{3} + \left(13 a + 8\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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