# Properties

 Label 1.21.6t1.a.a Dimension 1 Group $C_6$ Conductor $3 \cdot 7$ Root number not computed Frobenius-Schur indicator 0

# Learn more about

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $21= 3 \cdot 7$ Artin number field: Splitting field of 6.0.64827.1 defined by $f= x^{6} - x^{5} + 3 x^{4} + 5 x^{2} - 2 x + 1$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_6$ Parity: Odd Corresponding Dirichlet character: $$\chi_{21}(11,\cdot)$$ 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:
 $r_{ 1 }$ $=$ $9 a + 12 + \left(16 a + 27\right)\cdot 29 + \left(22 a + 9\right)\cdot 29^{2} + \left(21 a + 22\right)\cdot 29^{3} + 10\cdot 29^{4} +O\left(29^{ 5 }\right)$ $r_{ 2 }$ $=$ $18 a + 24 + \left(13 a + 19\right)\cdot 29 + \left(13 a + 8\right)\cdot 29^{2} + \left(10 a + 24\right)\cdot 29^{3} + \left(12 a + 16\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ $r_{ 3 }$ $=$ $13 a + 24 + \left(13 a + 23\right)\cdot 29 + \left(21 a + 18\right)\cdot 29^{2} + \left(21 a + 6\right)\cdot 29^{3} + \left(28 a + 25\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ $r_{ 4 }$ $=$ $20 a + 28 + \left(12 a + 12\right)\cdot 29 + \left(6 a + 19\right)\cdot 29^{2} + \left(7 a + 21\right)\cdot 29^{3} + \left(28 a + 21\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ $r_{ 5 }$ $=$ $11 a + 27 + \left(15 a + 11\right)\cdot 29 + \left(15 a + 4\right)\cdot 29^{2} + \left(18 a + 5\right)\cdot 29^{3} + \left(16 a + 10\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$ $r_{ 6 }$ $=$ $16 a + 2 + \left(15 a + 20\right)\cdot 29 + \left(7 a + 25\right)\cdot 29^{2} + \left(7 a + 6\right)\cdot 29^{3} + 2\cdot 29^{4} +O\left(29^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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