# Properties

 Label 1.301.6t1.i.a Dimension 1 Group $C_6$ Conductor $7 \cdot 43$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $301= 7 \cdot 43$ Artin number field: Splitting field of 6.0.190896307.1 defined by $f= x^{6} - x^{5} + 28 x^{4} - 19 x^{3} + 357 x^{2} - 100 x + 1847$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_6$ Parity: Odd Corresponding Dirichlet character: $$\chi_{301}(214,\cdot)$$ 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:
 $r_{ 1 }$ $=$ $21 a + 13 + \left(5 a + 22\right)\cdot 29 + \left(2 a + 27\right)\cdot 29^{2} + \left(20 a + 22\right)\cdot 29^{3} + \left(3 a + 2\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ $r_{ 2 }$ $=$ $21 a + 24 + \left(5 a + 13\right)\cdot 29 + \left(2 a + 11\right)\cdot 29^{2} + \left(20 a + 8\right)\cdot 29^{3} + \left(3 a + 26\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ $r_{ 3 }$ $=$ $21 a + 9 + \left(5 a + 10\right)\cdot 29 + \left(2 a + 25\right)\cdot 29^{2} + \left(20 a + 9\right)\cdot 29^{3} + \left(3 a + 2\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ $r_{ 4 }$ $=$ $8 a + 27 + \left(23 a + 17\right)\cdot 29 + \left(26 a + 1\right)\cdot 29^{2} + \left(8 a + 21\right)\cdot 29^{3} + 25 a\cdot 29^{4} +O\left(29^{ 5 }\right)$ $r_{ 5 }$ $=$ $8 a + 2 + \left(23 a + 1\right)\cdot 29 + \left(26 a + 4\right)\cdot 29^{2} + \left(8 a + 5\right)\cdot 29^{3} + \left(25 a + 1\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ $r_{ 6 }$ $=$ $8 a + 13 + \left(23 a + 21\right)\cdot 29 + \left(26 a + 16\right)\cdot 29^{2} + \left(8 a + 19\right)\cdot 29^{3} + \left(25 a + 24\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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