# Properties

 Label 1.133.6t1.j.a Dimension 1 Group $C_6$ Conductor $7 \cdot 19$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $133= 7 \cdot 19$ Artin number field: Splitting field of 6.0.44700103.1 defined by $f= x^{6} - x^{5} - 7 x^{4} - 5 x^{3} + 53 x^{2} + 97 x + 121$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_6$ Parity: Odd Corresponding Dirichlet character: $$\chi_{133}(83,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $x^{2} + 29 x + 3$
Roots:
 $r_{ 1 }$ $=$ $20 a + 23 + \left(25 a + 20\right)\cdot 31 + \left(17 a + 13\right)\cdot 31^{2} + \left(8 a + 15\right)\cdot 31^{3} + \left(27 a + 3\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 2 }$ $=$ $20 a + 15 + \left(25 a + 12\right)\cdot 31 + \left(17 a + 13\right)\cdot 31^{2} + \left(8 a + 17\right)\cdot 31^{3} + \left(27 a + 21\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 3 }$ $=$ $11 a + 24 + \left(5 a + 12\right)\cdot 31 + \left(13 a + 23\right)\cdot 31^{2} + \left(22 a + 16\right)\cdot 31^{3} + \left(3 a + 5\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 4 }$ $=$ $11 a + 20 + \left(5 a + 28\right)\cdot 31 + \left(13 a + 29\right)\cdot 31^{2} + \left(22 a + 13\right)\cdot 31^{3} + \left(3 a + 29\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 5 }$ $=$ $11 a + 1 + \left(5 a + 21\right)\cdot 31 + \left(13 a + 23\right)\cdot 31^{2} + \left(22 a + 14\right)\cdot 31^{3} + \left(3 a + 18\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 6 }$ $=$ $20 a + 11 + \left(25 a + 28\right)\cdot 31 + \left(17 a + 19\right)\cdot 31^{2} + \left(8 a + 14\right)\cdot 31^{3} + \left(27 a + 14\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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